Yep, output tubes are meant to run hot. I've measured 7591 and EL84 output types running north of 300 degrees F on the glass envelope in typical operation, even if it seems a little scary compared to the world of solid state components. In push-pull AB1 mode, even at idle (no signal) conditions, at 34 mA cathode current your 7591 tube plates are dissipating about 13 watts apiece and some of the stock Fisher gear ran the tubes much hotter that this. That's like having a 60W light bulb under the hood without even considering heat from the screens and other tubes and components, like power resistors.
Tubes lose most of their heat by radiant heat transfer from the plates through the glass envelopes, so most of the heat will warm up surrounding components and the air around them. As you say, air movement is how everything keeps cool enough to avoid damage to the components. Most often I've read that it's best for the tubes to draw air away from the them instead of blowing air on them to avoid thermal stress (when one side is cooler than the other). From what I've seen, most tube gear has worked well for decades without any forced air flow unless it's housed in a tight cabinet (like some consoles). That's because Fisher, Scott and others cut holes in the chassis (and the individual wooden cabinets), allowed for space around the output tubes, etc. to produce natural convection for sufficient air flow. I suppose, though, that added air flow might extend the life of tubes and components over time and certainly, if space is tight around components, a fan will help.
The only applications where I've seen designed-in forced air flow consistently applied is around transmitting tubes where operating conditions require constant cooling (even water cooling!) to keep things from melting down.
