I once transferred an SSD with a Linux Mint installation on it to another computer. It booted up without any issues whatsoever so I’d say it’s perfectly doable.
Jellyfin is free and open source. To me that’s always the preferred option. Plus, it works very nicely. Haven’t used Plex in a very long time but when I tried it, I didn’t like it.
If you just want stuff to work with minimal hassle, Linux Mint is an excellent choice. PopOS is also often recommended, especially for laptops. If you really want to get into the nitty gritty, something Arch based might fit the bill.