Most of the "legacy love" for D&D seems focused on AD&D 1e (tho there are grognards like me who still prefer BECMI, B/X, Holmes, or even the LBB's). I have heard/read a lot of derisive comments about 2e; but I always thought that, at least initially, 2e was a distillation of 1e + UA + whatever other mechanics improvements. I believe it had the longest run as the "currently supported version"; and some features, like specialty clerics, are still house-ruled even with later versions.
It may be worth bearing in mind that a lot of groups used bits of 2nd Edition in their ongoing 1st Edition campaigns, and may well have used various 2nd Edition supplements with their 1st Ed games. That probably contributed to its run as the longest supported version, and also means that there's probably a good bit more love out there than people really credit it with.
1. What was wrong with 2e that has earned so many haters?
IMO, the biggest thing was the neutering of the game in a failed attempt to satisfy BADD and the like - by removing Assassins and Half-orcs, by explicitly disallowing Evil characters, and by removing demons and devils from the core (and then renaming them).
The other big issue was that this was in some ways the anti-Gygax edition - this was the edition that TSR put out after they'd driven out GG.
Mechanically, 2nd Edition is okay, and very similar to 1st in most cases. With all the strengths and weaknesses that that implies.
(Oh, yes - the 2nd Ed DMG is also almost completely useless. In particular, all that good stuff from the 1st Ed appendices should have been retained - those are the best parts of that book, IMO, and it was near-criminal to produce a 2nd Ed DMG that replaced them with an utter vacuum of useful material.)
2. What was Right with 2e that got lost in the version shuffle?
Speciality priests was probably the single biggest thing. Eliminating the excesses of some of the later 1st Ed books (notably UA) was also a good move.
But the major enhancement of 2nd Edition was simply that it reorganised a lot of the material, and rewrote it into a much more approachable manner - had I moved from BD&D to 1st Ed, it's likely that I would have given up in disgust and no longer be a gamer.
The other big thing was of course the settings. In business terms, these were utter poison to TSR. However, in terms of ideas, they're absolutely great.
3. If 2e were used as the baseline for 5e, what would the end product look like?
That would be a mistake. The adoption of a consistent mechanic in the d20 system was absolutely a good thing, and although neither 3e nor 4e were perfect (or anywhere close), any step away from a consistent mechanic would be a retrograde step.