Any community you build inside a walled garden can be taken from you. I do think that is important to keep in mind.
The idealism of the internet in the 80s and 90s never could survive past the growth phase.
Both are doable with modern servers and clients thanks to v3 chathistory. Ergo (server) and Goguma (mobile client) in particular make this work very nicely.
What it does technically could be replicated by current technologies.
Emojis, video streaming, screen streaming.
Discoverability by the masses is a tough problem to solve because there is really no way to monetize it. Does Discord just rely on Nitro subscriptions?
It's a gold mine of data for things like market research, ad targeting/fingerprinting, and more recently AI training.
I then switched over to https://www.irccloud.com and it's such a improvement as I can use it on my phone, now I'm pretty active again and I kinda missed it.
Alternatively there's many bridges these days so you can use Matrix with one of the many apps supported.
IRCCloud does. Or if you want an open source stack: Soju as bouncer, Gamja as web interface, and Goguma as Android app. If you have a paid Sourcehut account you get Soju+Gamja hosted for you at https://chat.sr.ht/ .
I'm not sure The Lounge and Quassel support push notifications, but they otherwise fit your requirements.
You replied to my post where I said I switched to IRCCloud and the person asked for self hosted alternatives to it :P
Alternatively using a Soju[1] with Goguma[2] for phone and Gamja[3] for web could work; FWIW, I use Soju, but not the other two.
1: https://sr.ht/~emersion/soju/