Better multi-device support is huge. And history existing when you come back from not being connected.
There are IRC solutions to this but IIRC it's something silly like a client that stays connected 24/7 to manage your user, then another client that connects to that client for your actual interface.
Yes, Quassel and IRCCloud are exactly such solutions.
But that always has to happen in such distributed systems, Matrix has the same – your home node actually connects to the Matrix network, and you connect to that.
Even on XMPP you end up running an XMPP server somewhere which connects via federation to the actual network, and your clients just connect to that server.
There are IRC solutions to this but IIRC it's something silly like a client that stays connected 24/7 to manage your user, then another client that connects to that client for your actual interface.