im not saying we should ban mewtwo from gen1 rands, but maybe it could be a bit less optimized for the environment. maybe we could remove barrier and rest? ive been playing the format lately and am fine (75 games 72 gxe) but barrier/rest (especially rest) m2 is just a drag. there's currently no semi-consistent way to handle mewtwo unless i'm missing something big. it heals your para with rest, it stalls crit fishing and prevents full para fishing with rest, it outspeeds your boom user and barriers for free. barrier/amnesia with that speed means theres very little offensive counterplay one-on-one, let alone the (likely!) probability it gets free entry on something that can't bother it. you can try sleeping it if you're lucky to have one and have preserved it, but it can amnesia as you switch the sleeper in and yeah good luck having your grass types take a +2 blizz. last-mon mewtwo renders entire games irrelevant. no team preview and no chance of a non-setup set (im counting rest as setup and don't feel bad about doing so) makes this feel more overbearing / uncompetitive than gen7rb xern.
psychic, recover, blizzard, thunderbolt, amnesia at least doesn't lock it into one moveset every time (and still lets it be a monster!), but if more diversity is desired than that, you have self-destruct and thunder wave, and maybe hyper beam as a fringe option
As someone who has played a lot of Gen 1 Random Battles(peak 89 GXE and 1737 Elo, countless games played across tournaments and ladder), as well as Random Battles as a whole, I strongly disagree that the Barrier/Rest Mewtwo set is problematic. As a moderator of the Gen 1 Random Battles community, I think I can safely say that the Barrier/Rest Mewtwo set is generally seen as one of the worst, if not
the worst Mewtwo set one can roll... there's a few points you brought up that I'll touch on here. I will preface this by saying that all Mewtwo sets are definitely very good, I just disagree that it is a problematic presence within the format any more than, say, Gen 7 Slurpuff.
I will first note that, as you may already know: in Gen 1, the paralysis speed drop does not go away when a Pokemon rests it off until it switches out. In a format where Thunder Wave and Body Slam are so common, it is usually not difficult to get Mewtwo paralyzed, especially since a +2 Psychic OHKOs very few mons that are not weak to it. With the speed drop from Paralysis, this set becomes slightly less scary since you can re-paralyze it upon wake up with a twaver, or even put it to sleep, which can be very viable strategy since the Barrier/Rest set often functions as a "last resort, late game sweeper".
As far as the point about Blizzard, Mewtwo with Rest or Barrier will not ever run Blizzard, as if a Mewtwo has either Rest or Barrier you can be 100% sure that the set is Rest/Barrier/Psychic/Amnesia, so I don't really understand how that matters here.
As far as things that can "one vs one" the Barrier/Rest set, there are quite a few options. Mr. Mime can go in on it, Thunder Wave it, and then Seismic Toss it down, being able to stall it pretty effectively and likely knock it out eventually with the power of Paralysis RNG. Chansey also has some sets that can help deal with it, whether it be through TWave/Seismic Toss, freeze fishing with Ice Beam, Sing, or just general PP stall with the right support. Hypno/Drowzee also handle it extremely well, with the bulk to take boosted psychics easily and put Mewtwo to sleep, drop its speed with Thunder Wave, and even do solid damage with Seismic Toss(Seismic Toss in general is a great move to have to help deal with this set). Alakazam/Kadabra are 2 other strong choices to deal with Mewtwo when they run Seismic Toss. Persian, Meowth(as well as strong Slashers in general), and Raticate also help deal with this set, as they outspeed Mewtwo and can hit it super hard even with Barrier/Amnesia set up. Some of the counters I named may seem imperfect, or like they simply wall Mewtwo but struggle to take it down without some RNG... but that brings up a notable weakness of the Barrier/Rest set: PP. Psychic only has 16 PP, and since it is the only attacking move on this set, mons that can tank boosted Psychics and recover back the damage are able to potentially PP stall this Mewtwo set with proper support.
Pokemon is a game of odds management, and Gen 1 Random Battles represent that more than just about any other format. As annoying as it is to hear, some of the best ways to bring this Mewtwo set down if you don't have any of these Pokemon may involve freeze fishing, crit fishing(much higher crit rates in Gen 1 so this is actually quite valid), Paralysis fishing(which is much more effective than you let on above due to the speed drop not going away upon rest), and other RNG based methods. It's a hard Pokemon to deal with, but I can confidently say that it
usually does not autowin games for me or my opponent when it shows up. In fact, I'd rather roll a Mewtwo set with coverage in most situations, since it is much more versatile and consistent in any matchup, whereas the Barrier/Rest set needs a lot to go its way to work properly.
As far as your last point about adding Self-Destruct or Thunder Wave, I think these are interesting suggestions but I'm willing to bet most players of the format would agree that these simply make Mewtwo's current movepool worse and would not be desirable. Ultimately, Random Battles staff try to focus on optimizing sets as much as possible, and this would likely go against that.
TL;DR Barrier/Rest Mewtwo is a very good Pokemon but it is not the best Mewtwo set in the format, and Mewtwo itself is arguably not even the best Pokemon in the format due to the level nerfs that have already taken place to help balance it. Sometimes it can lead to some annoying reverse sweeps lategame, but that is just an inevitability of any main gen Random Battles format, and Barrier/Rest Mewtwo is no worse than many other examples such as Belly Drum Slurpuff in Gen 7. The set has multiple checks/counters or methods of counterplay to help play around it.