It was the last day of the great feast (the 8th day of the Feast of Tabernacles and hence a high day Sabbath or annual Sabbath) that Jesus “stood and cried out” (John 7:37) this uttering!
It indeed is true that the Holy Spirit was not given to the Public from the time of the Incarnation of the Word to the time of His glorification. That is roughly 33 plus or minus years. (Once earlier I had checked this and verified it).
John the Baptist filled with Holy Spirit in the womb was before the birth of Jesus. Same with Elizabeth and Zachariah! The prophet Simon was already filled with the Holy Spirit. He was “righteous and devout” (Luke 2:25). This shows clearly that he had received the Holy Spirit before Jesus’ birth.
The only activity of the Holy Spirit during Jesus’ earthly life was between God the Father and the Son of God. One example is the time of Jesus’ baptism! Other than between them, the Holy Spirit was inactive with the public world! (I am ready to be corrected if wrong).
So the question remains; why was the Holy Spirit not given to the world until Jesus was glorified?
The answer is simple (in my understanding):
There is only “one Spirit” (Eph 4:4; 2:18); “same Spirit” (1 Cor 12:4). There is no “another Spirit” (2 Cor 11:4).
Yet the Scripture calls the same Spirit as the “Spirit of God” and the “Spirit of Christ” (Rom 8:9) interchangeably!
So, if there is only one Spirit (no another Spirit) and the “same Spirit” is called the Spirit of God and the Spirit of Christ, there is only one conclusion:
The Holy Spirit proceeds from the Father through Christ!
In fact the Scripture states that all things are from God the Father through Christ our Lord:
“but to us there is one God, the Father, of whom are all
things, and we for Him, and one Lord Jesus Christ, through
whom are all things, and we by Him “ (1 Cor 8:6).
So Jesus describes the Holy Spirit as the Comforter “whom I will send to you from the Father” (John 15:26); because “all things which the Father has are Mine” (John 16:15). “And all My things are Yours (Father’s), and Yours are Mine” (John 17:10).
Thus it becomes clear that since God the Father creates and interacts with the humanity only through Jesus Christ, it was necessary for Jesus to be glorified so that the Holy Spirit can proceed through Him into the world! Since Holy Spirit is Spirit, to proceed through Christ, Christ needed to be Spirit. "The Lord is the Spirit" (2 Cor 3:17) after the Glorification!
This is why Jesus said:
"for if I do not go away, the Comforter will not come to you. But if I go, I will send Him to you" (John 16:7).
This is what John is witnessing in 7:38-39.