I've pondered the Woods Rifle/Walking Rifle a bit too, and compared it to what I call the GP Rifle as well.
What has been my walking rifle for a long while has been an early Marlin .357 lever, just as the OP suggested and Grump seconded. Loaded up, its pretty serious medicine for deer, maybe hogs and black bear as well. As it gains near 300 fps or more on similarly chambered handguns, I've no doubt on the SD role. The tube holds a bunch of beans, and more if loaded w/ hot .38's. Yes, it can take small game nicely, with single loaded .38 WC (won't feed in my rifle) but mild RN will and is nearly as efficient. I'd think its about like a .36 muzzle loader when so loaded. And you can buy all that ammo over the counter, without having to load.....well you used to anyhow.
The .357 lever does give up some range to "real" rifle cartridges, and traditional sights handicap it as well. A peep helps and a little scope more so, but you give up handiness and ease of carry if you scope one. Handy it is too. Mine is the lightest centerfire in the safe.
And while Coopers Scout Rifle concept, most recently manifested as the the Ruger Scout, comes to mind as the ultimate GP rifle, and I like Scouts mind you, ....I cannot place it into the Walking Rifle role. Too heavy, too noisy, kicks to much, and the GP caliber of .308 is too much cartrige for small game. Now with some sort of 100 gr Plinker, or a round ball over a modest charge, it can be loaded down, but I wonder if it will shoot anywhere near point of aim with full power ammo. My old Marlin will shoot 125 mag JHP and 148 .38 Spl WC to the same point, pretty useful.
As a side note, a recent competitor to the trusty Marlin has been a dandy .223 Mini Mauser with 6x scope (small bell). It almost carries as easily, though a tad longer and not as flat and compact, and the optic is there at the balance point. But I get more reach and accuracy, a lot more. No recoil, but a bit noisier. Ammo is still prolific, or was. I can load it heavy with Partitions, bonded or mono-alloy bullets and have a border line deer rifle. Everybody knows what a varminter they are loaded otherwise. I have not experimented with reduced loads, though it could be done down to Hornet levels, but I do not see the need.
The need, the reason the Mini Mauser had moved into that role is ol' Mr. Coyote, who frequently pops into ( and out) of sight out of range of the lever carbine.