Is “use to” a phrasal verb?

I know the answer seems pretty obvious, but I looked everywhere I could and found no answer. I found no reliable source clearly stating that “use to” is a phrasal verb. Allow me to explain to you why I am confused about this issue. Take the following sentence in the past simple:

“I used to eat watermelon.”

“Use to,” has only one use: to introduce past habits (which were lost, obviously).

The sentence above also makes it clear that “use to” is a transitive verb. But what kind of object is “eat” then?

The true source of my confusion is that it seems that “use to” takes the “to” away from the infinitive, probably to avoid repetition. If you tell me that “eat” in the sentence above is, indeed, in the infinitive, then I am at peace. It will be clear to me that “use to” is a phrasal verb. If not, does “to” belong to “eat” then? And if he latter is the case, is “use” always common verb and most people simply incorrectly write it together with “to?”

Thanks in advance.