I'm still not certain what you're thinking of in terms of a "general purpose rifle." I've read your original post several times, but am still not clear on the highest goal. Something capable of hunting deer-sized game, but fun to shoot, for someone who isn't generally an outdoorsman ...
If you want a GP rifle, it ought to be in a rifle cartridge. That rules out the pistol caliber carbines. I like my Marlin 1894 in .357, but I consider it a "fun gun," not a real hunting platform ... even though it is capable of taking deer-sized game. So if you want a lever action, your choices are pretty limited (.30-30, or .35 Remington) unless you go to a BLR.
If you start the conversation with "general purpose cartridge" (and in my opinion, one should always choose the cartridge first, then find a suitable launcher), I'd opt for a .308 Win because of its wide applicability in a short-length cartridge ... which makes carbine length guns possible. If you don't want the recoil of the .308, consider either the 7mm08 or the .260 Remington. Both of these are great intermediate cartidges that are big enough for deer-sized game, but not so big as to be tiring to shoot a lot. You might even drop to the .243 Win.
All of these outclass the .30-30 as "general purpose" cartridges in my opinion. But if you want a traditionally styled lever gun, you're back to .30-30 or .35 Rem as the choices ... and given that choice, I'd opt for the .30-30 due to availability, even though I prefer the .35 Remington as a dedicated hunting round.
FWIW, my "go to" general purpose hunting rifle is a Remington Model Seven in .308. I have it set in an aftermarket H.S. Precision sporter stock, mounted with a 2-7X compact Leupold scope in the conventional position over the action (NOT a forward mounted "scout" setup). The gun is light, points readily, is fast on target, and "accurate enough" (just at 1 MOA from a bench with inexpensive Remington Core-Lokt ammo). .308 is available in many different bullet weights and configurations, and is easily reloaded for if you want to go that route.
I'll second spending some time reading Cooper's The Art of the Rifle to get a good idea what a "general purpose" rifle is. But it's also "okay" to decide you don't really want what he defines as a GP rifle. If you like traditional lever actions, buy one and be happy with it. It isn't like this is a life commitment. If you find, eventually, it doesn't float your boat either, sell it for something else. For us hobbyists, "the hunt" is half the fun.
But you DO HAVE TO report back WITH PICTURES when you make your final determination!
Edit: And if you can find one (no, you can't have mine), an older Remington 760 Carbine in .308 is the cat's meow!