I don't see why this should be an adverb phrase.
Of course, the construction is highly conventionalised, probably even lexicalised and no longer productive (you probably couldn't say My car uses to malfuction a lot), but syntactically, this is a perfectly valid past tense verb which takes a verb in bare infinitive as an argument (this is why * used to malfunctioned or * used to malfunctioning is ungrammatical).
If it were an adverb phrase, what would be the finite verb in the sentence then? There needs to be one, and certainly this is used (to), due to its inflection, due to its syntactic position which is the usual default position for finite verbs and due to there not being any other possibilty to find a finite verb in the sentence, without which it would definitely be ungrammatical.
What would your arguments in favour of it being an adverb phrase be? I don't see any similarity with an AP here; used to isn't anything that modifies a averb, but is a verb itself; also, I know of no case where an AP could take a verb complex in bare infinitive as an argument (which used to does), in general adjunctions can not have any arguments (that malfunction is an obligatory argument of used to gets clear when we remove it and get an ungrammatical sentence: * My car used to); and most importantly, an adverb phrase would be merged into the syntax tree by adjunction, i.e. it would be an optional constituent which just modifies some other constituent (in case of adverbs, this is mostly the verbal phrase) but can easily be left out without this changing the grammaticality of the sentence (My car used to malfunction vs. * My car malfuction a lot).
This may hold true for a lot, but in the use of used to, this is definitely not the case, this is not just some adjunction that merely modifies its core constituents, but an a core contituent itself, namely the VP.
So I really don't think used to could in any way be analysed as an adverb phrase, and I don't see the problem of it being analysed as a finite verb, only because its use is in some way restricted and more or less lexicalised.
I must revise my claim that used to is not productive: I use to play tennis on modays is perfectly fine, and the fact that the full inflection paradigm is possible and regular (I use to, you use to, he uses to, ..., I used to, you used to, ...) I would see as another argument that used to is a totally normal finite verb, just in this context to some degree lexicalised/conventionalised and usually used in past tense.
The native speakers here have convinced me that my grammaticality judgement about present tense is probably wrong ;) So this doesn't seem to count as an argument for used to being a fully productive verb; but still, the other factors I mentioned are sufficient evidence in my opinion.