If privacy is a main goal, you generally can't get any more private than self-hosting. Do you plan on syncing while away from your home network? That would complicate things greatly simply due to the fact that you'd need to expose your self-hosted git instance to the Internet.
If you're fine with only syncing on your home network, using your old laptop, I don't really see any real security risks.
If you don't want to expose git directly, you can always expose a VPN server (wireguard?) to the Internet. This is the strategy I use. Then you can VPN into your home network anywhere you have an Internet connection, while still remaining relatively secure.
I've developed a couple of products used "in production" using 13" 10 year old sony vaio! It's definitely doable, but the experience is less than ideal, mostly due to the small screen. I don't think that's something a 15" display would fix though. If I were to do it again, I'd probably look for something around the 24" to 27" range.
Working with CAD on a 13" screen vs. a 15" screen is basically the same experience.