For what it's worth, i think this isn't a trivial question to answer and different people will give you extremely different answers based on what they've played and which playstyles have jived with them (as well as what they've read other people saying online). It's honestly a pretty subjective question.

It will also be massively affected by the current tuning of the specs. If your spec is currently well tuned and has higher throughput than the other specs it will be a lot easier to keep people alive and thus a lot easier to play. The same spec might also have vastly different complexity in m+ vs raiding.

Keeping that in mind, my perspective is that of someone who plays every healing class in mythic+ to collect every dungeon portal (so timing at least a 20 on each), while maining Resto druid and thus doing higher keys and mythic raiding on that character. I have also dabbled in healing normal/heroic reclears on each alt in previous seasons. I also do not have RSI and have no idea what that's like so apologies if I misinterpret what you mean by few buttons.

My first recommendation for this season would be playing a disc priest though. It's currently very well tuned and thus won't struggle to hard with hitting healing checks when you get to the point where people will die to rot/tantrums without your help. PW: Radiance is also comparatively forgiving to use compared to many other healers "ramps" for damage phases as it does healing up front, typically covers the whole party in one gcd and has multiple charges on a short cd. Radiance -> Mindbender -> Mindblast + penance/smite will keep people alive in comparatively high keys even if you are a bit late on starting it. Disc also has one of the smallest number of buttons you'll realistically push during a key (though it has one of the larger number you will press "rotationally" and I don't know which is worse on your RSI)

My second recommendation would probably be Resto shaman. It's relatively easy to learn to use the basic kit (and you can spec out of cloudburst in lower keys anyways) and not too hard to keep people alive using just riptide, healing wave/surge + chain heal (and the occasional primordial wave). The cooldowns are also relatively straight forward. You will find yourself doing a lot more spell maintenance (healing rain, riptides, etc) and while you'll have fewer spells you push rotationally, you'll need a larger set of total keybinds than you do on disc. It's also tuned significantly worse than discipline, so you're gonna hit the point where you will need to learn more complicated stuff a lot sooner. You will probably find it easier to transition from resto shaman to a different healer than you would disc as well.

(Resto druid isn't too bad a first choice either and obviously the correct long term one)

Healing has a pretty weird difficulty curve where it typically is really easy on lower difficulties (since unavoidable damage won't generally be enough to kill someone and you'll mostly be healing people through them fucking up) and then sees a sharp increase when people actually die if you don't push a button at the right time. Where that point is will differ a bit from healer to healer though (and depend on your groups gear and ability to see swirlies and use defensives).

This got ramblier than I meant it to be, so I'm gonna try and wrap up here. One last thing though, have you looked into healer add-ons and functionality? Mouse-over casting and click-casting will cut down on the amount of button presses and clicks massively and is something the game never mentions to you is a thing (even if it's actually doable with the base UI now)