Early shotguns were usually break-action. Bolt action and pump action shotguns came later, and for a while, bolt action shotguns were the better choice, just because early pump action shotguns had reliability issues. Pump actions would often bind up or would fail to properly eject the spent shell.
Mossberg was one of the first companies to sell a bolt action shotgun, and their line of bolt action shotguns was highly regarded at the time.
Pump action shotguns got better, though. They solved the binding and ejection issues, and pump action shotguns also started getting much more competitive from a pricing point of view. After WWII, leftover pump action shotguns from the military flooded the market. Mossberg still sold their line of bolt action shotguns (they ceased production during the war, but started up again afterwards), and other manufacturers continued to produce bolt action shotguns as well. You could even buy bolt action shotguns from Sears for a while. Some of the Sears versions had an unfortunate tendency to occasionally blow up in your face.
Once pump action shotguns solved their early reliability issues, hunters quickly realized that they could get off subsequent shots more quickly with a pump action. That, combined with the surplus of pump actions from WWII, led to the domination of pump action in the shotgun market.
As senoy said upthread, pump action is faster, but throws off your aim for the follow-up shots. Shotguns don’t need the precision aiming, and so favor speed over accuracy. Rifles favor precision over speed, so they naturally favor bolt action over pump, and those who want rifles with speed just go to semi-auto instead. I believe Remington is still producing at least one model of pump action rifle today.