Can the element in a phrasal verb have a syntactic purpose?
Phrasal verbs often have idiomatic combination and meaning that cannot be guessed from the verb alone, or from the adverb or preposition that follows.
In the idiomatic expression (that uses phrasal verbs),
He's been pushing up the daisies for a year.
it matters little whether the "up" in "push up" has a syntactic or semantic purpose.
Nevertheless a phrasal verb does often retain its "literal" meaning that can be guessed by its "elements".
In Longman Dictionary of Phrasal Verbs (Rosemary Courtney)
hold up a adv
- to raise (sth.)
"Hold up your hands so I can see them!" is meant literally.
to delay (sth. or sb.) [often passive]
to stop (a vehicle) by force (often during robbery)
Thus, "I don't really mean to hold you up, but I really need to borrow some money!" could be ambiguous without context.
AmE to charge (sb.) too much
to show or offer (sb. or sth.) as an example
to last
"Will my truck hold up through the winter?".
to remain in control of oneself
to show pride in oneself as in hold one's head up
to take someone by force or to threaten as in hold sb./sth. up to/for ransom
to make unkind fun of ; cause (sb. or sth.) to be the object of laughter as in hold sb./sth. to ridicule
Thus, you can only sensibly analyze the parts for definition 1. for the phrasal verb "hold up". Definitions 2 to 7 are disparate in meanings as in idioms. Definition 8 to 10 have their own idiomatic expressions.
By the way, Longman Dictionary of Phrasal Verbs (Rosemary Courtney) does classify phrasal verbs into 25 groups by grammar, but it would be too complicated for fresh ESL students.