On Discord, I open the app, go to a server, and can see all the rooms and all the messages almost instantly.

On Element (at least on Android), chats from different communities intermingle with my groups.

are you using the spaces feature in element? that's the same thing as discord "servers". they are on the left, unless you have none yet. the default setting in element is a bit silly, you should turn off showing rooms from all spaces when a space is not opened, it'll be much better.

on phone the space list is at the bottom.

I tapped on a large and slow-moving group, and watched messages slowly lurch into view as most of the messages were "join" and "leave" ones.

which app are you using? element X, or the old, plain element? the old app is slow, the new one should work much better in that regard.

you can also disable showing name change and membership change events if you don't care, but membership is good to be aware of and shouldn't be a problem with element x

Yeah no in no world would I call it perfect. The web version had problems of decrypting encryption keys without a phone also connected and syncing for me. I used to have an account with Element directly for a subscription so that I could have certain bridges. Since they discontinued this and botchered the transition they shut down that homeserver. I wasn't paying for a while but I expected to still be able to use the normal account matrix features on their homeserver. They just shut it down and mentioned it a few months previously in their blog posts but I didn't ever receive a mail or anything. Their data export/transfer tool stopped working with their home server shut down. I would argue actually for smaller instances because I don't trust the Matrix.org team as they split of from Elements but small servers also have a risk because they often depends on individual admins

That's emberrassing, I misread people talking in an issue about an open pr (https://github.com/cinnyapp/cinny/pull/2599) with them talking about an existing feature. That pr does seem reasonably close to landing though.

PWA? Fill a noob in.