I don't hunt big game (or deer

) but sure have allot of SKS, rimfire and .223 experience under my belt. While I've heard deer @ close range with 7.62x39 is a pretty good set-up, the accuracy limitations of both the ammo, AND guns like the SKS might result in a deer hunt not going your way. My buddies who use rifles for deer=.243/.270, and those are never long-range propositions either. The question needs to be asked whether or not the risk of a poorly placed shot (gun/caliber limitation~not the shooter necessarily) is one you can live with, or, if it makes more sense to get a heavier-hitter for the job. I can see where you're coming from with respect to the .223 logic, but in cases like this...I don't think there is 1 tool that will meet all the criteria you have.
A budget plinker~for sure, 22LR is the way to go. Those Cooeys are good (I've had at least 2-3 Cooey single shots) but I far prefer mag-fed bolt guns. I like CZ/BRNO/Anschutz most, but have put thousands of rounds through various Marlins/Savages/Henry/Chiappa/Taurus/Winchester/Norinco as well.
Scope~less than 10 years ago, you'd have had a hard time convincing me any scope was worth more than $150. lol I thought guys buying good scopes had deeper pockets, or were optics snobs. Well...that's true in allot of cases, but 4 things changed my opinion on that;
1. having several cheap scopes fail in a relatively short period of time (reticles flopping over, adjustment knobs that either don't function properly, or function at all, moisture finding it's way in in spite of never using on wet days, etc.)
2. my eyes aren't what they once were
3. shooting ANY gun allot will eventually reveal it's shortcomings, same goes for optics. Long shooting days=eye strain with the poor glass in cheap scopes
4. getting an opportunity to actually shoot guns wearing good scopes. My scopes (always well maintained) looked like someone had cleaned the outer edges of the lens with an oily rag. The good scopes looked like HD TV in comparison, and low-light visibility was staggeringly good.
I think it's the natural evolution of things, like fly fishing. At first you're so thrilled to be catching, you're happy to get 100 fish even if they're small. Big vest, cheap rod/reel, 10 fly boxes, thousands of flies, 8 hour days, etc. 30 years in, nice rod/reel, no vest, one fly box, maybe a dozen patterns, fish for the best 2-3 hours of the day, and only hunt for big trout...and totally happy even if you don't get one. lol (not that I ever get skunked

) I used to be happy with having multiple rifles, all equipped with the cheapest scopes I could get. Now, I'd rather have 1 rifle with a good (or very good) scope. I wouldn't suggest you have to spend a ton of money either, but once you decide a decent scope is a priority...believe me when I tell you that it'll actually elevate the shooting experience immensely. I was very stubborn about this for the longest time, because I never felt I could afford one. Then (thanks to a friend) got to try rifles equipped with Sightron SIIIs, US Optics, Swaro, Leupold, etc. Totally rewired my thinking on the subject, and it wasn't about bragging rights either.
Just my ramblings. The advice on "what rifle to get"~I'll leave for the deer hunters. I have personally seen how well the Savage Axis guns shoot, I've used .223 (x2) and .270 (x1). All 3 shot under 1" @ 100 yards with good (Hornady) factory ammo...and I'm no stellar shot either. All were the first generation of Savage Axis.