Suppose your adventuring party came into a situation like this...
[...]
Losing would lead to a bad result, like getting kicked out. My question is which edition would you want to do this in?
Quick answer: Use walkthrough, because such a silly situation can appear only in CRPG.

Serious answer: Encounter including poetry contest would
not look completely crazy in some settings, e.g. vaguely Japan or Arabian, or places where bard class is used widely. But if you run such a setting, you must know that this challenge happens, and so
already have mechanics. Perform(poetry) or "Poetry" proficiency from AD&D 2
The Complete Bard's Handbook... that's moot point: just one skill, not too hard to add or adopt if needed. More important is that either quickest, sneakiest or aesthetically perfect answer can be valued more, sometimes such rhyme would involve allusions - Innuendo, Knowledge(something).
I mean, even though one variant resemble another,
ultimately it's setting-depended thing, and even in one setting it's situation-depended. No guarantee of single-check resolution, and rulebooks would not give all possible variants. So -
Final answer: ideal variant is something that combines checks better than 3ed; otherwise indiffirent.
I would make the characters play it out in-character, regardless of the edition.
Well, of course it's best played, not rolled, but just like with diplomacy skills, basic mechanics adds uncertainty and helps to avoid going OOC. Why not tell rhyme and
then make check, to determine audience reaction ? Or why not to
adjust roll depending on request details, much like in other situations ?
What fun would something called a "poetry competition" be, if it could be resolved by just another throw of the dice?
I agree: by the same token,
“oh, your PC searched chap's room as quickly as possible and found this and this” is less satisfying than
“<my PC> looks under bed” - “nothing but chamber-pot, it stinks.” - “carefully tilts mirror's top and look what's behind or stuck on backside...”- “there's some parchment, it's folded”. First approach
is surefire hazard-killer. But this does not means second should not involve any roll, right ?..