As others have said, their relative non-popularity is a quirk of the couple legacy items of game makeup (1. horses have been incredibly fragile, don't scale up with level, and most of what make them necessary eventually can be replaced by bags of holding; 2. any build-mechanics or treasure-selections dedicated to mounts turn off in at least some common environments). The things contributing to their popularity would probably be 1. they are an iconic fantasy element, and 2. people like pets.
Basically, D&D has done everything it could for a long time to make them unattractive.
Indeed. And also did its' best to give you alternatives to having them. Perhaps another reason was that a lot of the advantages also were a lot better in
Chainmail than in oD&D/AD&D. Being really good at besting plate armor is good and all, but if most of your opponents are armorless monsters*, maybe it doesn't matter as much. Double damage is good, but if you spend any rounds** lining it up or anything it quickly stops being a go-to option if the expensive*** thing that facilitate it might drop out from under you.
*who might need magic weapons to fight, in which case (because of the magic item chart) you want longsword proficiency
**or just second attacks, once you hit the level for that
***and if you used training costs, you are cash-strapped in AD&D
It's remarkable how the concept has held on despite the stereotype that the horse will die quickly.
Well, there are plenty of people who might want to play a knight on horseback. It's just a dream that dies quickly for many if the system doesn't support it.
It is an interesting space, things that no small portion of gamers might want to play, but the game historically hasn't done well with the implementation. The best contra-example I can think of is TSR-era Thieves -- a roguish skulker-type is a popular concept,
but the game made them unfun to play*,
butx2 people still did play them (quite a bit, if the anecdotes I hear are representative). I think the primary difference is that both the game and some play assumptions many made conspire to make playing without mounts easy (bags of holding and flying carpets and gauntlets of 'carry 300lbs extra' are common, lots of people glossed over some part of the wilderness adventure minigame, etc.), whereas they kinda conspired after supplement I/AD&D to make Thieves feel necessary (counter to the pre-thief assumption those who played from the very very beginning would have experienced) -- or at least lots of people interpreted the presence of Thieves as indicating only they could do some things and that those things showed up in common dungeons.
*exceedingly fragile, rather bad at things other than your primary role, low chance of success during the dungeon-centric part of the game, if you do fail you're often subject to the trap or all the people you were trying to sneak past/up to
Hilariously, 2E had a whole section on mounts, including a table of horse traits.
I'd love to see mounts (and classes tied to their use) be much more viable in the game and the system simply more friendly to their use without turning them into pocketable pokemon like 3E tried to do.
2E was rather famous for a 'if we include a bunch of options and focus on the flavor, people will use them in spite of us not changing the underlying game system that disincentivizes such'. The
Complete Fighter's supplement had rapiers and sabres cutlasses and belaying pins and martial arts -- but you were still going to be up against undead and golems and demons that needed +1 weapons to hit and the same magic item table that made longsword the right choice. The
Complete Thieves' supplement had dog pepper and hand warming lamps and tar paper that added 1-5% (or just removed penalties DMs hadn't known to be subtracting until these were introduced) to thieving skills --which didn't change that the best way to play a successful thief was to convince your DM that now was not the time to make a check. And so on (pacifist priest options, spell-less campaigns, stone and bone-wielding neolithic options -- all without really examining how this would work in the D&D frame). I don't remember the horse traits table (was it in
Complete Paladin, I think I stopped collecting by then), but a big old table of horse traits without any rules changes to make them more playable seems very on-brand for 2e.
As for pokemon-horses, I think that's what 5e has done/is doing for pets in general (along with making them come back after a specific recharge if slain). Just making them things that might get left behind if you exit the dungeon not-where-you-entered, or just things that die, makes them really more beneficial for someone who treats them as an expendable resource compared to someone who treats them as an exciting addition to the PC's characterization, and I think they are leaning into the later.