Can a used Mac Pro 6,1 running Catalina run an older version of Final Cut Pro?

You could get an older version but inly if you had already purchased the application.

You could use a more recent Mac and make a purchase using YOUR Apple id, and then download the older one on your Mac. But you’d miss out on the newer features and performance. Maybe iMovie will do what you need, and it’s free. Or if you need more, some people here swear by DaVinci Resolve, and there’s a free version too. I don’t know if they still offer an older version but it appears to be less demanding in terms of its OS version requirements.

Please try downloading the trial version that you can use for free for 90 days without purchasing FCP all of a sudden. Once the expiration date has passed, you can no longer use it. If you like it, please buy it.

Tangentially have you considered the free copy of DaVinci Resolve which is every bit as good as FCP minus the Voice Isolation and Magic Mask?

With Monterey you could download Resolve 18.6.6

https://www.blackmagicdesign.com/support/family/davinci-resolve-and-fusion