Depends on the type of deer hunting your doing. I've used a mates Remington pump 30/06 when stalking deer & the forearm part of the stock rattles just when you don't want it to. I've used the same gun flushing pigs & have found it good for follow up shots.
I've been on two shooting trips where we were chasing pigs on motorbikes & my mates pump failed due to the dust, where the bolt actions the rest of us had continued to work. These were however extreme conditions lasting over a week on both occassions.
But really, whatever rocks your boat. I've tended to find after many years of deer hunting that its usually the first shot that counts, rather than the ability to fire quick successive shots, thus negating the influence of action used.