My Alphas do. My Omegas sometimes do, sometimes use Marauders. A battlecruiser is perfect for running down any high sec anomaly up to and including Dens, and also Level 2 or Level 3 missions. A Marauder is much better for Level 4 missions.
I do wish people would stop saying that we are suggesting “changing the battlecruiser’s role”. That role is defined solely by what people use it for. Nowhere is there a gospel according to CCP that it is not intended for PVE. No one is demanding a change in cargo, simply suggesting that the logic of ship class progression means that perhaps it should have a better cargo hold. It is staggering how arrogant, childish and aggressive people are over such observations, even suggesting that proponents are trying to break the game.