Let me layout the occurrence of the verbs used to send "Christ":
- Christ sent by an unspecified authority (the Father implied??), Matt 10:40, 15:24, 21:37 (parable), Mark 9:37, 12:6, Luke 4:43, 9:48, 10:16, John 7:28
- Christ sent by the "Spirit of the Lord", Luke 4:18
- Christ sent by "God", John 3:17, 6:29, 8:42, Acts 3:20, 26, 1 John 4:9, 10
- Christ sent by "the Father", John 5:36, 38, 6:57, 10:36, 11:42, 17:3, 8, 18, 21, 23, 25, 20:21, 1 John 4:14.
- Christ sent by an unspecified authority (the Father implied??), Luke 20:13 (parable), John 4:34, 5:24, 30, 6:38, 39, 7:16, 18, 28, 33, 8:26, 29, 9:4, 12:44, 45, 13:20, 15:21, 16:5
- Christ sent by the Spirit, --
- Christ sent by "God", Rom 8:3
- Christ sent by "the Father", John 5:23, 37, 44, 8:16, 18, 12:49, 14:24
Thus, we may deduce that, indeed, John is the only NT writer to explicitly specify "the Father" as the one who sent Jesus. All other writers simply say either "God", leave it implied, or, "the Spirit of God". John also uses most of these expressions as well.
Note that John also uses the idea of Jesus being sent more often that all other NT writers. Further the verbs ἀποστέλλω (apostelló) and πέμπω (pempó) appear to used almost interchangeably, sometimes in adjacent verses.
Interestingly, John 5:23 offers compelling evidence of the equality of Jesus and the Father: that all may honor the Son just as they honor the Father. Whoever does not honor the Son does not honor the Father, who sent him. Ellicot succinctly observes:
The purpose of the entire commission of judgment to the Son, a
bestowment which illustrates the quickening results that he (who does
the will of the Father) wills to effect, is now gathered to a lofty
climax, abundantly vindicating the right he had claimed to call God
his own Father. It is as follows, in order that all may honour the
Son. Τιμῶσιν, not προσκυνῶσιν ("honour," not "worship"), is the word
used; but seeing that the identical sentiment of reverence due to the
Supreme Being, to the Father, is that which is here said to be due to
the Son, and is here declared to be the reason why all judgment is
entrusted to the issues of his will, - we are at a loss to know how
loftier attributes could be ascribed to the Son.
I agree. Perhaps John was at pains to make things more explicit to counter the (then) rising tide various isms that deprecated the status of Jesus. John ensures that Jesus is not just God (John 1:1-3), but "the God" (ho theos), John 20:28; about which Ellicott observes:
My Lord and my God.--These words are preceded by "said unto him," and are followed by "because thou hast seen Me, thou hast believed;"
and the words "my Lord" can only be referred to Christ. (Comp. John
20:13.) The sentence cannot therefore, without violence to the
context, be taken as an exclamation addressed to God, and is to be
understood in the natural meaning of a confession by the Apostle that
his Lord was also God.