[xPaw]

The Lounge is an IRC client that solves the problems you describe (persistence on any device, synchronised unread counters, push notifications) - https://github.com/thelounge/thelounge

It does require having your own server to host it on though.

[linuxftw]

Hadn't seen this before. Looks nice. I tried the demo, smooth other than the /j is not an alias for /join, which I think is an obvious oversight.