I'm constantly worried that I didn't get enough sleep, so I'm hoping to prove to myself that that is not the case. How could I do that without sending all of my health-related data to Google or some other company?

I hate to be a Debbie Downer here, but my advice would be to not use anything to track your sleep. I used an app to track my sleep for about 12 years, and the number of times I thought "I feel great today!" - but the app said my sleep was terrible (and vice versa) was so high that I gave up on it altogether.

I've also been reading more and more reports of other people seeing similar results with Oura rings, Apple watches, etc lately. Seems like it's all just junk science.

I've noticed that it's pretty much useless when I tries to calculate your sleep quality. But it does do an ok job keeping track of the total time you slept.

Garmin watches with Gadgetbridge, and if you're on GrapheneOS you can even deny it network privileges.