In one respect, you have already answered the first two parts of your own question within the question itself:
God ... is referred to as "father" (as in Matthew 6:9, Romans 1:7, Ephesians 5:20, 2 Thessalonians 1:1)
And this is correct. However, "they name" is merely a reference to Πάτερ (Pater), since the sentence states "Father who is in heaven, your name is sacred (hallowed)." As such, "thy name" should be understood simply as a second-person pronoun referring to the noun in the first clause of the sentence. In other words, if anything is going to be understood as being an attempt to avoid "using the of God in vain" then it would be the use of "Pater," (as opposed to Theos or Kyrios) not "your name".
Despite this, the reason for using this phrase has nothing to do with Exodus 20:7
What's in a name?
The way the OP's question is phrased betrays a misunderstanding as to what the author of Exodus meant by "taking the Lord's name in vain." In fact, if the use of πατήρ (Pater) were an attempt to avoid the use of the Divine Name, this would have more to do with Exodus 3:14.
Throughout antiquity, many ancient peoples had a concept of a Divine or Secret Name. This is best illustrated in the form of a legend about the Egyptian god and goddess Ra and Isis. In this legend, Ra becomes injured, and Isis uses this fact as leverage to learn the divine name of Ra. Isis tells Ra that she could only heal him if she knew his secret name. Isis immediately cured Ra, but he could not take back the power that he had granted her by telling her his true name and from that point on Isis was equal even to the sun god in power.
It was believed in most mesopotamian cultures in antiquity that a god's true divine name contained power and that by learning that divine name an individual could control a god and gain power over that god. Therefore, most ancient spells spells and incantations involved some wording along the lines of "By the name of [divine name] I command [action]" - because it was believed that this lent the spell power. For example, on page 124 of Jewish Aramaic Curse texts from Late-Antique Mesopotamia by Dan Levene we see a spell in which the canter is instructed to use the name of Hadriel and Shakniel to silence "evil and violent people who stand gainst Berik-Yeheba son of Mama"
In the name of Hadriel, Shakniel, the well, the stone, and the pit, I adjure, I adjure you, in the name of he who is great and frightful, that you may silence from Berik-Yehaba son of Mama the mouth of all the people who write books, who sit in forts, who sit in market places and in streets, and who go out on the roads.
Another on page 46 seems to utilize as many names as possible as a power-enhancement tactic for the spell
I have adjured you by the holy angels, and by the name of Metatron the pure angel, Nidrel and Nuriel and Huriel and Sasgabiel and Hapkiel and Mehapkiel, shose seven angels that are going adn overturning the heavens and the earth and the stars and the zodiac signs and the moon and Plaedes. May you go and overturn evil sorceries and powerful magical acts...
Understanding the power of a Divine Name, it becomes much clearer what is going on in this scene: Moses is fishing for God's divine name and hoping that perhaps Yahweh is one of his lesser names in an attempt to control God. Instead of giving it, God answers with a name similar to his actual name in much the same way Ra answers Isis with his lesser names (notice God's answer to Moses in Exodus 3:14 is only one letter of difference from the spelling of Yahweh). Instead of giving his name, God responds by saying "I am who I am and I will be what I will be." With this one pithy response, God has both answered Moses and signaled to him that he will not be controlled by any mere mortal and that no use of his Divine Name (which Moses already knows; Yahweh) will control him.
You will note from this however, that the use of Πάτερ (Pater) avoids the actual use of the divine name and ensures that this would not be a concern.
Ancient Mesopotamian Contract Law
Within Mesopotamian contract law, many individuals in antiquity attempt to leverage the concept of a divine name to attach a curse clause to contracts as a penalty for the violation of said contract. For example, in Marriage as a Covenant: Biblical Law and Ethics as Developed from Malachi Dr. By Gordon P. Hugenberger of Boston's famed Park Street Church, Dr. Hugenberger gives some examples and notes:
...a considerable number of marriage contrats do include an oath. For instance, eight of the forty-five neo- and late-Bayloanian marriage contracts assembled by M.T. Roth invoke a curse against anyone who would violate the terms of the agreement...
...curses are similarly placed after the stipulations and immediately before the list of human witnesses. Although there is significant variety in detail, the curse in No. 5 is typical (lines 26-29):
"May Marduk and Zarpanitu decree the destruction of whoever contravenes this matter, and may Nabu, the scribe of Esagila, cut short his long days. May Nebuchadnezzar, king of Babylon, decree his destruction."
Considering the presence of oaths and curses in contracts which already have human guarantors, D.L. Magnetti notes that while
"contracts were made in a sphere in which men could take care of the situation... the fact remains that evidence indicates that oaths were sworn as part of contract procedure in at least some ancient Near Eastern civilizations. Perhaps this was due to influence by the procedure in the law court [where oaths of clearance or oaths for witnesses were required at times] or to a desire for the additional sanction of the supernatural"91
91 D.L. Magnetti, "The Oath in the Old Testament," 49f
This then seems to be what Exodus 20:7 is oriented towards: it is meant to prohibit this kind of behavior for many reasons. For example, God is encouraging Israel to be a people marked by their integrity and honesty. This can be illustrated in the sermon on the mount in Matthew 5:33-37
“Again, you have heard that it was said to an older generation, ‘Do not break an oath, but fulfill your vows to the Lord.’ But I say to you, do not take oaths at all—not by heaven, because it is the throne of God, not by earth, because it is his footstool, and not by Jerusalem, because it is the city of the great King. Do not take an oath by your head, because you are not able to make one hair white or black. Let your word be ‘Yes, yes’ or ‘No, no.’ More than this is from the evil one.
Here, as with preceding subjects in the sermon, Jesus is referring directly to the commandments given to Moses. Jesus is encouraging his disciples to be men of integrity who needn't invoke the divine and add curse clauses to their contracts to bolster their reputation.
Similarly, this reflects back on the God of Israel. What would happen if one of the contracted added a curse clause and Yahweh chose not to curse that individual upon violation of his contract? And what would it say if the the people of Israel frequently entered into contracts including penalties from Yahweh and then frequently broke the terms of their agreements? What kind of respect of and fear of their God would this demonstrate to other groups?
This then is the real meaning of "not taking the name of the Lord in vain," and in fact, the Hebrew word שָׁוְא has an alternate translation of "falsehood" which would render the text
You shall not take [use] the name of the Lord your God for falsehoods, for the Lord will
not hold guiltless anyone who takes [uses] his name for deceptions or falsehoods.
The NET notes also hint at this stating:
שָׁוְא (shav’, “vain”) describes “unreality.” The command prohibits use of the name for any idle, frivolous, or insincere purpose (S. R. Driver, Exodus, 196). This would include perjury, pagan incantations, or idle talk. The name is to be treated with reverence and respect because it is the name of the holy God.
What about Jewish Customs?
One final thing to note is that many would (rightly) point out that it is rabbinical tradition not to speak the name of God. But as Tracy R. Ruich notes,
Jews do not casually write any Name of God. This practice does not come from the commandment not to take the Lord's Name in vain, as many suppose. In Jewish thought, that commandment refers solely to oath-taking, and is a prohibition against swearing by God's Name falsely or frivolously...
Nothing in the Torah prohibits a person from pronouncing the Name of God. Indeed, it is evident from scripture that God's Name was pronounced routinely. ...The Name was pronounced as part of daily services in the Temple.
The Mishnah confirms that there was no prohibition against pronouncing The Name in ancient times. In fact, the Mishnah recommends using God's Name as a routine greeting to a fellow Jew. Berakhot 9:5. However, by the time of the Talmud, it was the custom to use substitute Names for God. Some rabbis asserted that a person who pronounces YHVH according to its letters (instead of using a substitute) has no place in the World to Come, and should be put to death. Instead of pronouncing the four-letter Name, we usually substitute the Name "Adonai," or simply say "Ha-Shem" (lit. The Name).
So as we can see, at the time of Jesus (and Matthew 6:9) and until at least the time of the writing of the Mishnah (circa 217 CE) it was not thought to be prohibited to speak the name of God. This did not arise until some time between the Mishnah and the Talmud (circa 500 CE).
Based on the references you cited (Matthew 6:9, Romans 1:7, Ephesians 5:20, 2 Thessalonians 1:1) as well as many other clues of context, when the Gospels were composed, the author of Matthew would have have undoubtedly expected his audience to understand God's name to be Yahweh.
But why did the Author of Matthew choose to use Πάτερ (Pater) instead of "Theos" or "Kyrios"? It seems we can infer this from two simple and obvious possible conclusions. The first is that it makes a theological statement. The term "Father" (Πάτερ [Pater]) is one of Respect, but familiarity and intimacy. It's use makes a theological statement that Yahweh is a relational God who desires to be intimately involved in our lives in a way that could be compared to a fatherly role.
The second possible reason for this is the same reason for using any new term: the existing terminology is not adequately specific or does not adequately describe or embody what the speaker or author is trying to convey. While I know that the OP is not a Trinitarian theologian, under this paradigm, it would be difficult to distinguish between Theos the father, Theos the son and Theos the Holy Spirit. Using Πάτερ (Pater) would be a more succinct and simpler way to distinguish between the members of the Godhead and be a more effective term for communicating the author's meaning (assuming the author of Matthew was aware of the concept of and believed in a Triune God.)