Does your mobile IRC client notify you (push message) when you're mentioned or receive a private message, but only when you're idle or offline on your desktop client?
I looked in to it after that answer. Colloquy[1] was the top hit for my search, and yeah, it does it, but (of course) you've got to have a bouncer[2] running all the time and connected to any channels you want to monitor. Probably pretty nice once it's set up, assuming there's a way to run it on a cheap VM or Raspi or something rather than your workstation or laptop. Looks like their mobile client attempts to register its device ID with bouncers in a channel when it connects, so that's automatic. Not exactly a competitor with what Slack and similar are doing—not having to set up and manage this sort of thing to achieve those services' features is exactly why people pay them—but seems like a nice solution if you prefer IRC.
A well-tended IRC server has those via bots.
Other important missing features are ease of use in terms of creation of
> topic-focused channels
wut? This is totally easy in IRC: /join #topic-channel makes the channel
> private channels
I'm tempted to say that a bot can provide that... but sure.
> and high quality mobile and desktop clients.
Not buying that, sorry, I've been happy with my mobile/desktop clients; they are relatively full-featured.