Old Sap, you must have never seen anyone proficient with a bolt. A bolt used to be an infantry rifle and there have been some very fast people with bolt rifles through the years. My favorite off the shelf bolt action will consistently shoot under 6" at 1K yards.
You have seen some shooters manipulate a bolt action quickly - but this does not translate into the bolt action being quick to fire, especially with any accuracy. Having to let go of, then later re-engage the trigger is a major factor in the four separate motions necessary in order to shoot a follow-up shot with a bolt action. This does not lead to a combination of speed - and accuracy.
One would seriously doubt that, with shooters being equal, anybody could keep up with the rate of fire - and accuracy while doing it - of a good pump with any bolt action rifle ever made.
Why is quick, accurate fire important? - Sometimes the game animal needs a second shot, right then and there - and sometimes they do not pose for you but instead are rapidly heading into thick brush. In such a situation, a good sportsman is more concerned about a quick, accurate follow-up shot than he is about stunt shooting at 1,000 yards - and rightly so.
A sportsman puts the animal first, realizing that good intentions does not always lead to a one-shot kill, in much the same way that elaborate and certain battle plans do not always hold up in the fog of battle.
Since most game is taken at 100 yards or less and not at 1,000, there is a lot to be said for the responsible hunter considering a design that will give him the fastest, most accurate follow-up shot possible at those ranges, and that design is not the bolt-action rifle, which is only marginally quicker for fast, accurate fire than a good single-shot.
Then there is reliability to consider. If an autoloader jams after the first shot, then the second shot will be mighty slow in coming. And, as we all know, autoloaders jam more often than any other design, period. So, once again putting the animal first, what you want is the ability to reliably deliver a second shot if necessary, as quickly and accurately as possible, at ranges most often encountered in hunting situations.
The surest route to that is either the lever or pump action rifle.
I use a BLR in .308 Winchester - but that's just me... I will readily acknowledge that the pump aficionados here have the facts on their side.