783's are accurate guns for sure, had a couple, one in 7mm one in 30'06, the synthetic stock hunting versions, both really sucked up the recoil and shot quite well, I forget what I traded the 7mm for (had too many) but I kept the '06, bedded it in a Boyds stock and it was night and day, made it an exceptional shooter. I'd still go the Savage route vs 783 personally, Savage has a ridiculous bolted together bolt design, but the bolt guide acts like a gas seal, and I'd put it right up there with a 700 for gas handling.
This is a $400 Caballas 12VF in 6.5CM, threw it in a $500 MDT stock, it's now a worthy competition rifle, there are a lot of ways you can go for under $1000 to get something respectable.
This was some load development for he 6.5, left group is 120gr PRVI bullets, it hates them. Middle is FL sized 140gr Nosler Custom Comp, right is Redding NS with same load as the FL sized, clearly this one like 140's and Neck Sizing. The bull is about .75".