Having grown up with a .30-30 as a "woods gun" decades ago I will give my interpretation of what that term means.
A woods gun means a rifle that you are going to carry all day through thick brush and trees. Shots are going to be almost always 25 - 75 yds. You are typically hunting deer or black bear size game. This is the typical of hunting in the forests of western Washington and Oregon. I hear it may also be typical of some of the dense woods in the southeastern states.
What this means is that fine accuracy is not a priority. It should be light and easily carried in one hand for hours at a time. You don't need or want a lot of doo-dads attached to it as anything sticking out will get hung up on branches, vines, etc. Also, it needs to be heavy enough caliber for medium size game.
For all these reasons the .30-30 lever action rifle has always been popular. It is slim, slick sided, and balances perfectly in the hand which can easily wrap around the receiver. Some guys rely on it entirely. Others will have a .30-30 when they know they will be hunting through the dense woods and crowded draws, and a scoped bolt action like a .243 or .30-06 when they know they'll encounter large clear cuts or more open woods on ridge tops.
If I were to use an AR rifle for dense woods I would keep it as clean as possible to avoid hangups on brush. Iron sights, and I would think seriously about whether I would put a sling on it or not. Usually a woods gun is going to be in one of your hands while the other is pushing branches out of the way.
None of the cartridges you mention are overkill for the woods. It is the handiness of the rifle that matters. If you get the r-25 platform then you have the same size rifle regardless of the cartridge. So you might as well go for the cartridge most appropriate for the game you hunt. Any of them would be fine for deer. However, if you go for bear or boar then you probably want to 7-08 or .308.
By the way, the .243 cannot be overkill if you think the .257 Roberts is good. The .243 has more velocity but a smaller diameter and weight bullet. I would rate them about equivalent for woods range.
For a lighter "woods gun" you might consider one of the larger caliber AR15 versions. The 6.8 or 6.5 cartridges would all be pretty good up to deer. If you like them, then for the short ranges in the deep woods, the .458 and .50 cartridges for the AR15 might be fun. If there is a 10mm upper then that would certainly be interesting, too. This would be similar to using a .44 mag version of the lever action instead of the .30-30.
From what you describe I would think the AR15 in 6.5 Grendel would be a better woods rifle than a larger one in the .308 family.