I thought a more advanced distro, that is still stable, would be good overall. However, not getting new software for a long time sounds quite annoying.

Arch is actually not as bad as many say. It's pretty stable nowadays, I even run Arch on some servers and I never had any issues. It gives you the benefits that you can basically find any package in the AUR and everything is up-to-date. Try it out, if you don't like it, you can still switch to something else.

I’m wanting to challenge myself to get much better with Linux, partitioning, CLI, CLI tools

The best way to learn the CLI is to use it. Try not to use your graphical file manager for a while and only interact with the file system through the terminal, that teaches you a lot.

I’ve been considering installing Arch the traditional way, on my X220, as a way to force myself to improve. Is this a good way to learn more about Linux and a Linux system in general?

Yes.

I always hear good things about the Arch Wiki.

It is truly fantastic.

Is there any other tips someone can give me, to sharpen my Linux skills?

Use the system, don't be shy, try different things out. If you are scared that you might break something, try it out in a VM. Break your VM and try to fix it. That teaches you a lot.

I was even considering trying out Gentoo on my X220, but the compiling times sound painful.

I would not recommend that, updating packages will take ages, it's not a great experience.