You probably mean Xbox One X, not Series X, which didn't come out until 2020. The console sold today for $500 is a different system, it's more powerful and has newer games.
We need to know exactly what's wrong with the console and what the repair shop wants to charge for, with as much detail as possible. If it's something that doesn't require advanced repair skills, maybe you can do it yourself and save the $300, or at least a large part of it.
If the old one was connected to the internet I think it will have possibly done cloud saves. I’ll defer to someone else as I had the original Xbox, then went towards PlayStation until this generation. Though I did have an Xbox one for about a week before returning it to get a ps4 a few years ago. So my series s is my 1st Xbox in a while.