Weirdly I feel like they used to switch me back to "Home" constantly but it doesn't seem to happen anymore. It only changes when I change it now.

When using Tweetdeck, never.

https://tweetdeck.twitter.com/

If you like Tweetdeck you should try it with the browser extension Better Tweetdeck.

Seems to be the default if you log out and back in. Which I do a lot. But otherwise it (seems to) stick(s).

I remember this happening on Facebook but it hasn't happened to me on Twitter that I can remember. I've had the setting on for at least a couple years without issues.