In "мне захотелось пить" - should I think that past tense «захотелось» means «A desire to drink has come to me, and as a result there's an ongoing state where I want to have a drink now»?
Although it may mean that the desire has come or had come depending on the context, I won't use the phrase to say I want to drink now.
To make it sound natural in the present perfect sense you have to relax it, make it sound more colloquial, for example:
Мне что-то захотелось пить. - I'm kind of thirsty.
Notice that что-то is not an object of пить, but just a word used to mince the phrase.