Most maps will be aimed at running for between 15 and 30 minutes. Under your proposal, that would be a maximum of 30 seconds worth of firey fun per map. Hardly worth the effort when you consider just how complex it is to create a flamethrower that's more realistic than a giant blowtorch.
I think it's worth mentioning at this point though, that most flamethrowers from the middle of the war onwards used a cartridge system for ignition rather than a pilot light or electrical trigger, as those were too unreliable. Those cartridges were contained within a revolver mechanism and were effectively bullet blanks i.e. everything except the actual projectile. These were much more effective, but they restricted the number of times you could actually pull the trigger, because a full tank of fuel is useless if you've no longer got anything to light it with.
By limiting the number of times you can actually press fire, that changes things dramatically, because you no longer have an ubersoldat running around squirting a .25 second burst at everyone he sees for instant barbecue kills. They instead have to carefully ration their cartridges against their fuel, choosing when and when not to use it. A natural balancing mechanism if you like.
Of course, whether Stalingrad era flamethrowers used that system yet, I honestly don't know, but I believe that they probably did.
