Here is the thing that sometimes people do not realize: riding without a saddle is actively harder on the horse.
Think of it as the difference between using a really well designed long distance camping, hiking or infantry backpack, vs just having someone dump all that shit in a bag hanging off your shoulders.
Now the Noldor fucking love horses. The Noldor are big on horses. The Fëanárioni shipped across as many horses as they could in the open-decked swan-ships, Maedhros apologized to his uncle (and thus by proxy to the rest of the Indisioni Exiles) by gifting them horses, and the strong implication is that Fingon essentially gleefully lived the entire Siege of Angband as a ferocious horse nomad.
Likewise, they fought on horse-back and while it is not impossible to fight on horseback without stirrups, it's much harder and much more likely to hurt both you and the horse.
The Noldor and the Rohirrim have the same attitude, about horses.
They have also had horses for thousands of years, learned horse-care from the creators of horses, and so on. Glorfindel's first round through life was as part of those born-in-Aman Noldorin Exiles.
So of fucking course the Noldor have saddles. Similarly, of course they have reins: reins are how you communicate with a horse whose head is way the fuck over there. (As is pressure from your knees, and so on.) Now, none of these are of the coercive kind, and none are the kind that would use discomfort as a communication way, but none of those are necessary anyway.
On the other hand, the Nandor of Mirkwood-once-Greenwood . . .don't seem to have any significant equestrian history.
And why would they? They are intensely forest-dwelling people. Horses are not naturally forest creatures, and in particularly dense forests do not provide a significant transportation advantage - especially not when being ridden. A horse in Mirkwood in a battle is mostly a liability - you can see echoes of the same thing with the Rohirrim in both their relationship to Fangorn, and in their intense apprehension at getting the guidance of the Druédain through the woods to get around that one orc-host during the journey to Minas Tirith.
Sure, sure, stories of deep woods monsters and so on, but also part of the reason that the forest is part of the Rohirric Cultural Imaginary as a Terror Place is that their one major military strength doesn't work there. They're a cavalry-based fighting people, and deep tangled forest is no place for cavalry.
Horses, to get through woods, need roads. And the Nandor of Mirkwood don't seem to be big on roads. They only even seem to have roads when interaction with other cultures demands it; they prefer using rivers and using their own feet in the forest. When trade with the outside world decays because it gets dangerous, the One Road through their kingdom also decays and is abandoned. And even if you do keep around a few very sure-footed little ponies for baggage, because they're pretty good at that part and can keep their feet through the trees, well . . . you don't RIDE those. And frankly they're more likely to keep donkeys.
Legolas' people do not appear to have significant traditions of riding horses and particularly not for battle.
But you know what specifically the royal, Sindar-origin line of the Greenwood has a history of?
Pride, and being massive show-offs, sometimes in ways that get them into deep trouble.
This is how Legolas' grandfather died, and how his father became king: during the Last Alliance, Oropher got it so up his craw and his neck so out of joint about actually following the directions of Those Obnoxious Eldar and Númenórean Snobs* that he and his compatriot Nandor king charged very prematurely and got themselves and their people slaughtered, basically To Prove That They Couldn't Be Bossed Around/Did So Know What They Were Doing.
[*please note: these Eldar and Númenórean snobs had been fighting wars - significantly against Sauron, and in the case of the leadership quite personally, and in the case of one member of the leadership against fucking Morgoth - for several thousand years and were intensely good at it. Oropher did not, as far as is recorded anywhere, even participate, let alone lead, in any significant military campaign.]
Like don't get me wrong, I'm deeply fond of them, but also this is a thing they do.
Similarly, we know that Legolas personally is both a showoff and gets his nose a wee bit out of joint when he feels miffed or insulted. Gandalf has to tell him and Gimli to stop sniping at each other, and at more than two thousand years old, Legolas has a lot less excuse for getting involved in petty fights than most anyone else in the party.
When they're snowed in on Caradhras, Legolas expends no small amount of energy in exerting himself to run across the snow for relatively little gain - but damn does it make him look impressive when he gets there! He is VERY touchy about the idea that he should be treated like everyone else in the party (ie as an outsider) when they're trying to get through to Lórien, and later feels the need to make sidelong comments about feeling young, as he hasn't "since travelling with you children" when they're on the edge of Fangorn, and so on.
And right up to that point, Legolas has had a couple of unpleasant experiences - the hobbits were kidnapped and Boromir was killed and the three of them haven't been able to do anything in particular about it; Aragorn has been much more materially USEFUL in the chase than he has, even if he himself might have been able to run without stopping more; and just now this bunch of humans insulted an ally, implied insult to his entire species, and threatened to kill his friend, and then Aragorn resolved the whole issue by being DIPLOMATIC about it instead of anything else - and is surrounded by a bunch of mortal Atani.
I put it to you: might it not be a major temptation to show off?
Because to be clear, while yes Glorfindel has saddle and bridle, he is able to verbally instruct Asfaloth to ignore Frodo pulling on that bridle and run, from a couple meters away. I don't think the idea that the Quendi can get horses (and other positively inclined animals) to do what they want regardless of external measures of control is at all inaccurate. Glorfindel has a saddle because a saddle makes Asfaloth more comfortable carrying his weight (and is terribly convenient for storage and baggage), and has a bridle because laying reins across the neck is very useful way to communicate without making noise, which has all sorts of advantages.
But Legolas - given that his family trades extensively with the Atani on their eastern border - certainly knows that Atani find the way that Quendi can just naturally ~*communicate*~ with animals and get them to do what they want very impressive; and also likely knows, from the same source, that riding bareback is considered an indicator of great skill.
Also, critically: as he does not ride horses OFTEN, he may have no idea how to put a saddle on, take it off, ditto a bridle, how to care for them, how to care for the horse after having them on, and so on, and in order to learn this in their current situation he would have to ask Aragorn and there isn't really any way he could hide that incompetence from Gimli, who is certainly now his friend but is also someone he still wants to impress.
Finally, practically speaking, depending on the exact design it might well be very difficult to keep Gimli on with him if the saddle remained.
So what better way to do that, and to overawe these belligerent Atani (who insulted his friend AND his people AND the Lady of Lórien AND mutter mutter muttermutter humans being stupid muttermutter), than to pull off this great trick?
Bonus: since he's always going to have to stop to let Gimli off before they fight, he doesn't need to worry that much about staying on DURING combat, because he won't be fighting that way.
Additionally, if you take the framing premise of the book (that it is written out of the recollections of the hobbits, primarily Frodo and Sam but with some additions from Merry and Pippin who are the other major points of view we have) seriously, this is a bit that would have been added in afterwards, and you can actually see that reflected in the language used (it is a LOT more High Register, throughout the sequence with "the three hunters", than it is anywhere that it's from Merry or Pippin or Sam or Frodo's primary point of view), very possibly Gimli's or Legolas' own, or some combination, suggesting a possible origin for the claim about "way with all good beasts" as Legolas would want to maintain his image, and Gimli's sense of what Quendi can and can't and do and don't do would be primarily shaped by, well, Legolas.
TLDR: The Noldor and Eldar in general of course use (very well made and perfectly fitted/suited to the horse and rider etc) saddles and reins to ride, because they're actually horse-cultures. They don't necessarily "need" them, and can in fact communicate with and convince horses to do things via other means, but saddles are for the horse's comfort as much as the rider's, and reins are just a SMART way to communicate in a rider context.
Legolas is not from a horse-culture and is not particularly accustomed to riding horses but figures it can't be that hard if humans can do it (and does have enough unfair elven advantage to pull that off) and is a massive showoff, and therefore made a big deal about not needing a saddle etc etc.
Aragorn let him do this because it wasn't that important (the horse was strong enough to carry both of them, this was not the time) and frankly it was kinda funny.
The text has this minor confusion because it's being written by hobbits who are outside of both cultures and lack significant context, were compiling a massive BEAST of a volume based on multiple contributors, and so on.
postscript: @lireavue absolutely reblogged this in order to trigger this post, and don't let her pretend otherwise.
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