痛い is a word that originates from Proto-Japonic > Old Japanese, so its roots go back farther than the existence of kanji in Japanese.
The reading of い from 医 is onyomi, meaning it only entered Japanese after kanji were imported to the language from Chinese.
Additionally, たい only means "want" when attached to a verb stem. It doesn't attach to nouns like that.
There's no connection.
To add onto what Leebo said, the radicals break down to "use" 用 and "carbuncle/puss-filled sac" 疔 (which does have a radical for sickness in it).
Without doing the research (because I also don't know the actual root), it looks to be related to a disease where they have to painfully pop large, thick-skinned pimples.
That said, you should try and research it using the Chinese reading ツウ. Let me know what you find!
You're looking for the etymology; the "where did the meaning the kanji 痛 come from"
Does the word 痛い come from 医 and たい?
I mean, sometimes you gotta ignore kanji because there are cases where they use different kanji but pronounced the same with the simaler meaning. So it got me thinking whether or not 痛い comes from a combination of both, 医 meaning medicine/healing and たい expressing 'wanting'.
And if it is true, what's the history behind it?