I believe that the best explanation of Jesus' baptism may be that He was being ordained as High Priest and this was "the fulfillment of righteousness." Hebrews presents Jesus as the ultimate High Priest, superior to the Levitical priesthood established in the Old Testament and in order to start a priestly ministry a Levite would be ordained at the age of 30.
In the Old Testament, at the age of 30 Levites who were qualified to become priests were washed and anointed for their service (Exodus 29:7). For example, Mosses washed or "baptized" Aaron and his sons and then anointed them with oil.
Lev 8:6 And Moses brought Aaron and his sons, and washed them with water,
7 And put upon him the coat, and girded him with a girdle, and clothed him with the robe, and put the Ephod on him, which he girded with the broidered girdle of the Ephod, and bound it unto him therewith.
8 And he put the breastplate thereon, and put in the breastplate the Urim and the Thummim.
9 Also he put the miter upon his head, and put upon the miter on the forefront the golden plate, and the [a]holy crown, as the Lord hath commanded Moses.
10 (Now Moses had taken the anointing oil, and anointed the [b]Tabernacle, and all that was therein, and sanctified them,
11 And sprinkled thereof upon the altar seven times, and anointed the altar and all his instruments, and the laver, and his foot, to sanctify them.)
12 And he poured of the anointing oil upon Aaron’s head, and anointed him to sanctify him.
Similarly, Jesus' baptism and the anointing of the Holy Spirit can be interpreted as the ordination of His priestly ministry.
We know that Jesus is the High Priest from our reading in Hebrews 5:1-6, where Jesus is depicted as a high priest appointed by God.
5 For every high priest chosen from among men is appointed to act on behalf of men in relation to God, to offer gifts and sacrifices for sins. 2 He can deal gently with the ignorant and wayward, since he himself is beset with weakness. 3 Because of this he is obligated to offer sacrifice for his own sins just as he does for those of the people. 4 And no one takes this honor for himself, but only when called by God, just as Aaron was.
5 So also Christ did not exalt himself to be made a high priest, but was appointed by him who said to him,
“You are my Son,
today I have begotten you”;
Melchizedekian Priesthood: Hebrews emphasizes Jesus' priesthood in the order of Melchizedek, a priestly order separate from the Levitical priesthood. Melchizedek was a priest-king who blessed Abraham in Genesis 14:18-20. Hebrews connects Jesus' priesthood to Melchizedek, highlighting its superiority over the Levitical priesthood.
Sinless High Priest: According to Hebrews 4:15, Jesus is described as a high priest who is "without sin." Unlike the Levitical priests who had to offer sacrifices for their own sins before offering for the people, Jesus, as the sinless High Priest, offered Himself as the perfect sacrifice once and for all (Hebrews 7:27).
John the Baptist, whose father was a serving priest and whose mother was a "daughter of Aaron" would establish John as a priest. John washes Jesus and then God does the anointing. Jesus then begins His ministry as the true High Priest.