Look into Gadgetbridge, as that allows the bypass of proprietary apps for some trackers

Garmin watch will track you with it's own GPS and doesn't need to sync with a phone.

Some of these suggestions are the over the top for what you need. Just get a Garmin Forerunner and don't sync it with your phone. They are easy to find secondhand if you want to save money and can be connected to a PC via USB without the need for a Garmin account.

I think the others didn't get it, but possibly it's just me, so let's make it clear: are you searching for fitness tracker hardware, like a smartwatch/band/whatever, or for software that interacts with that device?

For the former, I don't have any recommendations.
For the latter, try out gadgetbridge and fitotrack. Both are on f-droid

I do it the old school way and just use stopwatches. Especially with running, it's easy to just go by time and record it manually.

While this may not be what you're looking for, it's worth mentioning that a good ol' pencil and paper does wonders. It won't have everything you need, but you can time how long you ran for with a stopwatch, count how many pushups you do, manually measure your pulse, etc. If you're good with data processing you can stick the data in a spreadsheet and process it to see your progress. The bonus is you'll learn a lot more about health through doing it yourself. Besides that, I've never used a smart watch or fitness tracker. I've just exercised until I get tired.

I'm not sure to get it, sorry if it is not what you've been looking for but would this https://github.com/SamR1/FitTrackee work ?