I think you would to reassess your definition of ‘alliteration’.
You write: “… pitcher/spring (the p is the alliteration)”.
If a single repeated letter – in different positions inside a term, too! - would have be able to trigger an alliteration almost all the Hebrew Bible texts would cause alliterations to follow one another…
A couple of examples of clearer alliterations in the Bible:
Psalm 55:9 (מסער סעה), “storm, tempest”, with the repeated sound 'so-'.
Genesis 24:21-22 (משׁתאה […] לשׁתות), “gazing […] drinking”, with the repeated sound '-št-'.
In the instance you quote (Qoeleth 12:6-7) I see no alliterations, unless you define an alliteration the swap between the two graphemes ghimel and mem in the opening words of the verse 5 (מגבה גם), “Also, of heights…” [really, these two terms appear in the inverse order. I am not able to write them in the correct order, since the system swap them, independently. Sorry].
‘Is NIV taking poetic license?’
‘Is the original Hebrew poetic?’
Yes, but not owing to a supposed use of alliteration, but through the use of the poetic device of antonomasia. This term (drawing from the Greek verb αντονομαζω, ‘to change the name’) focuses on the literary device that imply that an expression takes the place of a proper name.
So, any meaning we may attach to the expressions we find in the text at issue (the explanations vary) we are before to some expressions that substitute the proper names of several body parts.
A single correspondence - on which all commentators agree - that we may find is in the verse 3. In fact we find the expression הטחות (‘the grinders’) instead of the proper name שׁנים (‘teeth’).
I hope these information will be useful for you.
P.S. Sorry for my wobbly English.