For dogs have compassed me: the assembly of the wicked have inclosed me: they pierced my hands and my feet. [Psalm 22:16 KJV]
One of the biggest objections to this verse not being used to refer to the crucifixion of Jesus is that it is never quoted in the New Testament. Isaiah 53 is quoted in various places, Zechariah 12:10 (looking on the one who was pierced) as well as other parts of Psalm 22 but not verse 16.
One answer I've come to think as somewhat likely is that the gospel audiences would have already been very familiar with this verse. The gospel authors then chose to emphasize other lesser known aspects of the messianic prophecies. For example, Matthew makes allusions to other aspects of Psalm 22 like Jesus' garments being divided, his trust in God, onlookers wagging their head at him but not the piercing of Jesus' hands and feet.
Something doesn't seem right about this interpretation but I also can't think of any arguments against it.