I have several shotguns that could fill that role. I have an older 870 Express with a 21" turkey barrel that takes choke tubes as well as a 20 cylinder bore with rifle sights. I can keep all my shots on a paper plate at 50 yards with just the bead on the turkey barrel. Can do it at 100 yards with the other barrel. I prefer the turkey barrel with a modified tube with buckshot for HD. But they stay home when hunting deer, hog or bear.
In many parts of the south deer are driven with dogs. I can see possibilities using a shotgun with buckshot for this type of hunting. But for what I do I think there are better choices.
Thick brush is where I do most of my hunting. Needing to quickly get off a shot and then needing rapid repeat shots is extremely rare. It just doesn't happen very often and when it does the chances of a hit are extremely rare regardless of what you're using.
A far more likely scenario is to just see an ear or tail flicker in a small opening in the brush. You almost never see the entire animal. You're just looking for a small baseball sized opening in the brush to thread a bullet through to hit a vital organ on a stationary, or slow-moving animal that doesn't know you're there. And it's going to almost always be within the 1st 5-10 minutes or last 5-10 minutes of daylight where it is near impossible to even see iron sights.
I've found a short, lightweight extremely accurate rifle with optics to see the small openings in poor light to be the best brush gun. The idea is to miss all of the brush and shoot THROUGH small openings which means precision shooting even at 50 yards. Even a 12 ga slug will deflect if it hits brush.
I cut the barrel to 19" on this Tikka 308. It weighs 7 lbs as shown. Even with the scope it is lighter and faster handling than my 870 with a 20 or 21" barrel. It is extremely accurate and with the scope I can see to make shot far earlier, and later in the day than would be possible with irons or dot sights. No, I can't get off a 2nd or 3rd shot as fast, but you'd be surprised at how close I can come to matching a pump shotgun.