Is there something similar for TikTok, that would simplify the interface, give you notifications for a few channels (if that's the right word), with privacy in mind, and all without the need to register there?

This is the solution I’ve been looking for, thanks for sharing it!

There's also https://fraidyc.at/ which provides a browser extension. It's an easy way to do exactly what the OP was doing and for more than just twitter.