In 1957 (or so) I shot a rifle for the first time. It was a 1912, 22 cal Remington model 12, take down similar to yours and I was about 5 y/o. As I remember we only shot it a few times before it quit working. It was never fired again until today when I shot it at 50 yds. I'd recently restored it, ordering parts from Numrich and then waiting several months for Covid quarantines to end, finally getting them last week.
The restoration was for my education and experience, more so than any added value, but it came out sweet. Despite having a very rusted bore when I looked at it carefully for the first time in over half a century, it cleaned up well, and shot amazingly well.
I refinished the wood, blued the steel where it'd been worn off from use, and as I mentioned, replaced a broken firing pin and other parts. I also purchased a new barrel for it and may do the swap later, just for the experience.
Your stock is likely repairable, maybe even available in good used, original condition. Rust is easily removed via chemical or electrolysis methods, and new or original parts may be available online.
I say give it a go! After all, what's the worst that can happen?