Which one would you recommend nowadays, PuTTY or (the fork) KiTTY?

If you have a recent version of Windows, the copy of OpenSSH you get with Bash for Windows would be ideal, as you then use the same tools everyone else does eliminating the Putty specific setup and terminal emulation bugs Windows devs generally need to go through. If you really need to use it, stick with Putty, as even then you benefit from other people having the same setup steps and experience, and as such common issues are more likely to be resolved.

Side thought: why doesn’t Putty follow a more normal setup and configuration process, i.e. why does it have its own private key format and tooling around that instead of using standards?

I couldn't find a terminal emulator I liked for WSL so now I use PuTTY SSHed to localhost to login to WSL...

You might like wsltty (https://github.com/mintty/wsltty). It's mintty for WSL which in essence is putty with just the terminal capabilites.