I'd be interested to know if anyone actually seriously uses something like this. I can't imagine there is much crossover in the irc audience and the people who would want to use a web based client. Even less so when you need to self host it.
I'm involved in a lot of open source but had used IRC only intermittently until I got The Lounge running.
I've really enjoyed it. I'm able to log in and see the same unreads and direct messages across my work PC, personal PC, and phone and iPad. The software has been reliable and relatively painless to update. The UI functions nicely on touch devices.
Before The Lounge I operated ZNC in EC2, but configuring device-specific clients to use it was a chore, and I didn't find myself using IRC very much. Now, I use IRC all the time.
Obviously if you're not comfortable maintaining a server then this isn't the option for you, but as a "power user" I've found the web UI perfectly tolerable. Instead of maintaining IRC clients on multiple devices and pointing them at ZNC, I maintain a single IRC client -- The Lounge.
Though paying money to receive messages while offline (or fiddling with ZNC) is one of the reasons IRC getting its lunch eaten by everything.
It's our default web-based client that we offer for public use over at EsperNet (which as an IRC network has been around since the 90s) - https://webchat.esper.net.
The fact it is modern, easy-to-use and has better security characteristics made it a bit of a no-brainer to switch from Iris.
It seems worth it for me to throw this on a DO droplet, give it a URL and totally forget about having missed something again. Right now I just use Textual with all the default configs installed and don't do anything fancy. For a casual IRC user who doesn't want to go into much more effort than that to preserve history or have seamless client experience/settings and am willing to pay 5 bucks a month to host in the cloud somewhere, I think I'm the target audience here.