It's actually incredibly easy to replace a watch battery on most watches (certainly Timex), you just need a tool to pull the back off ($10ish on Amazon right now) and a battery that fits, which you can pick up at any Wal-Mart for $4-$5.
This was valuable to learn because I since replaced the Timex with a Garmin Forerunner 220 (now 5 years old) that's much more expensive. The rechargeable battery now only lasts for about 2 hours of GPS activity or 5 days of non-GPS use, about half what it did when I bought it originally, so I'm considering an upgrade again.
Worth trying to fix perhaps?
Likely not a “good” use of your time, but fixing stuff is strangely satisfying. You seem skilled, but if you are just learning, then try to fix stuff that you would otherwise throw out and that doesn’t need expensive parts. It does “waste” time, but many people get a lot out of it.
I am confident I could replace the battery, less so that I could do so non-destructively and make it waterproof when I reinstall everything - it's not designed to be particularly repairable, you can see they had to cut the plastic to get into the strap in that video. It would be fun to try to write new firmware for the watch and add some of the Fenix features to the Forerunner.