We hunt whitetails in exactly the same terrain. Most everyone here prefers a short barreled, compact bolt rifle in an extremely accurate, flat shooting cartridge. The 308 is probably number one, but 243 and 7-08 are common and the new 6.5Creedmoor is getting some use. In 45 years of hunting I've seen 2 hunters carrying 30-30's and a handful carrying shotguns, all back in the 1970's. None in recent years.
You're only getting one meaningful shot and there is no such thing as shooting through brush. The best bet is an accurate, flat shooting rifle that will thread bullets through baseball size holes in the brush. Good optics that work in low light help you see those openings, and even at 50 yards the arched trajectory of a 30-30 can be a problem. I zero my rifles at 100 yards, the bullets are no more than 1/2" above or below my crosshair between 50 yards and about 130 yards making it a lot easier to hit the game and avoid brush.
Needing a 2nd or 3rd shot is very rare, but even if you do a bolt gun is fast enough if you learn how to shoot one. I can get off 3 shots in under 2 seconds with my bolt guns. And if you're talking about "aimed" repeat shots there is no difference between any of the actions. Put up a 9" paper plate and fire 3 shots as fast as possible with the requirement that all 3 must hit the plate. There will be no difference between any action type. It takes longer to get the sights back on target than to work any of the actions.