I believe the ONLY 2 semi-autos available in "deer calibers" at $500 or less would be a used Remington 740s, 742s or 7400s as well as an SKS. I have had dozens of Remingtons come through my shops in the last 45 years or so. (I have been a gunsmith now for 42 years)
These rifles get mixed reviews. Some are accurate and trouble free, but others are not very accurate and jam a lot.
You take your chances with them, and what's more, Remington’s customer service is legendary as being almost non-existent. So if you get a bad one……….oh well..!
Now the SKS on the other hand is ugly and fires a shell that is about like a 30-30, but they shoot pretty well, and are super rugged and reliable as a rule. If you are hunting at 200 yds and less, I’d take the SKS over the Remingtons even if both were offered to me for free. The SKS falls down if you need more power than a 7.62X39 can offer.
In sporting (non-military) rifles for hunting I’d put the Browning BARs at the top, but you are not likely to touch one for $500. Good used ones go for about $750 around here. Not all of them are super accurate either, but most are quite good. 2 MOA or a bit less is average from them. If you get one that’s not accurate Browning is a good company to work with in its customer service. They take good care of you.