Lvl 3 dispensers actually heal a little slower than a medics normal heal rate, 20/sec vs 24/sec
why - why would you finish off players with the shotgun THE FLAMES ARE ALREADY SCORCHING THEM THEY ARE ALREADY DEAD
Even more than what Bal said about damage, when I am playing pyro, I'll often lite up 3 people at a time and I can't only hold one of those people to the flames for long enough to kill them outright. At that point, unless the other two weer really caught off guard or are slower classes, they are running, running clear of flamethrower range. This is when the shotgun becomes the best thing for the pyro to have, you already have the advantage of your initial fire damage plus any burn damage to come, so 40-80 damage per shot depending on how much spray hits them is really all you need to come out the champion.
Also, while shotgunning, you can dodge around and not take everything in the face very easily. This is not really the case when you are (w + m1)ing unless you can get a good circle strafe going on a slow class.
Really, the main victim of my post burn shotgunnery: Medics. Doctors have 105% movespeed from the base, so chasing after them with flame thrower if you didn't manage to backburnner them into the ground from the get go is often a losing proposistion. You can still get them like that, their only ranged weapon is fairly shitty, but you can kill them a lot faster with two good shotgun blasts after your initial roasting. Reserve the Axetinguisher for slowpokes like Soldiers and Heavies that you can't get to the back of to back burn due to being spotted or being blocked out by other bodies on his team