7-08 or 243 would be my choice. ‘06 ruining too much meat is stupid but I get it, their rules ((shot placements like EVERYTHING ELSE, being the key) center punch the shoulder with anything listed and let me know how much is not damaged)). The bigger knock on it, imo, would be potentially introducing bad habits into a new shooter - ‘06 takes some shoulder. I have no idea how big your son is, but my narrow ass wasn’t ready for the -06 I used as a child, and it took a lot of shooting to unfuck some of those habits later.
Imo, there is still a felt difference between a 500/800/1500 gun, but that’s not to say there aren’t a TON of good options in the lower-middle range now that used to not exist. Get an $800 gun (Tikka/Bergara/etc) and put $750-1000 Leupold VX5 on it and keep for next 40 years. Personally I think you need a 243 (or 3) before a 7-08 but both have killed many, many, many deer. The kids on our lease all shoot 7-08’s, every last one of them.
Eta: weight does factor into the cost, but not at that price point. Also, a “good weight” depends on what you’re going to be doing. Strictly mountain hunting elk on long stalks, people will pay huge $$ to save those ounces. A young kid that can hump around a slightly heavier rifle, not as big a consideration. A heavier rifle will generally help you reduce felt recoil, although there’s exceptions both ways.