Is there by chance a simulator of this on Steam if a good beginner hobby kit doesn't exist?
Buy some cheap movements off eBay, and dive in. Yeah, you'll lose parts and break things but you'll learn a bunch.
I have a couple Westclox watches from the 60s that I bought for $10/each on eBay, and they both now run at +/- 2 seconds a day.
The screwdrivers and tweezers should be of good quality though, at least get the ones that cost 3-4$ each from China.
And you don't even need a mechanical watch. Cheap quartz watches also have tiny interesting parts in them that you can try to disassemble and reassemble.