The New International Version is in perfect harmony with Jesus' quoting from Psalm 8:2 in Matthew 21:16:
"Do you hear what these children are saying" they asked him. "Yes," replied Jesus, "have you never read, "'From the lips of children and infants you have ordained praise'?"
"From the lips of children and infants you have ordained praise" (Psalm 8:2)
As for the English Standard Version, it makes this comment against Psalm 8:2:
"The Greek translation of the Septuagint (see Matthew 21:16) rightly interprets "strength" as "strength attributed to God in song," or praise."
The children (in Matthew 21:16) were praising Jesus by cryinging out "Hosanna to the Son of David." The ESV comment says this:
"Jesus acknowledges the children's praise and links it to Psalm 8:2, which the religious leaders should have known applied such praise to God, thus confirming Jesus as the divine Messiah."
The New Living Translation also helps to shed some light on this. In Matthew 21:15 it says that the children were shouting, “Praise God for the Son of David.” Matthew 21:16 says:
”Haven’t you ever read the Scriptures? For they say, ‘You have taught children and infants to give you praise.” Psalm 8:2 in the NLT says:
”You have taught children and infants to tell of your strength, silencing your enemies and all who oppose you.”
The NLT Footnote explains that the Greek version reads “to give you praise” and refers to Matthew 21:16. The Hebrew root words for “Hosanna” are found in Psalm 118:25, which says, “Save us, we pray, O LORD!” (ESV). Literally, hosanna means “I beg you to save!” or “please deliver us!” In saying, “Hosanna!” the people were crying out for salvation, and that’s exactly why Jesus had come.
There is nothing here for Protestants to reconcile. Jesus did not misquote Psalm 8:2.