A few weeks ago I shut down my self hosted znc bouncer that I still logged in every few weeks to download logs, see if I got pinged somewhere and catch up with some low traffic channels.
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.
For IRC bouncers there's many, but I'm not aware of anything that has a web interface, phone app, push notifications and all kept in sync. The market is pretty small.
Alternatively there's many bridges these days so you can use Matrix with one of the many apps supported.
> I'm not aware of anything that has a web interface, phone app, push notifications and all kept in sync.
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.
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.