There's nothing to be gained by making the other classes more viable. Sure, you theoretically *could* figure up a way to make the other 5 classes strong enough to warrant them being run all the time, but the simple fact of the matter is that, that flies in the face of 7+ years of 6v6. At this stage, the game is what it is and radical changes would simply result in something qualitatively different from "6s."
Does Pyro need some kind of a rework for its current state? Yes. Although, in the realm of 6v6 I would say it is primarily a graphics related one (pyro flames make it damn hard to see anything in a way they never did before).
There have been many players who ran pyro in situations other than last defenses, Ma3la, Enigma, Huey, TMP, Satan, and many others. In fact, back in the day, it was fairly normal to bring a pyro to badlands mid (comes out of shithouse/clubhouse) at least once in a longer game.
Heavy also used to be brought out a LOT more than he is now. Hell, Oplaid go heavy was an invite team ;) Heavy used to come out all the time. Dheroes even wrote it into our DNA with "Dumbbrain Go Heavy!" any time we started to lose momentum. Check out an IM Vod from s13, to the open finals from s12 - you will see heavy on a non-last point *at least* once in a match if not for minutes at a time. People simply don't do it anymore because they're too skittish of the ire such a tactic brings, and because if they tried it in scrims they'd likely see the other team RQ. Enigma's heavy on Warmfront mid single-handedly ended that map's multiple season run in ESEA.
Sniper is already quite viable in a 6s setting so long as your sniper is actually good. That's the hard part. If you have a guy like Powah who can just shred people on command, then sniper can be brought out in just about any situation. The trouble is that most of the people who try to main sniper in 6s are bad at it and thus bias their cohorts against sniper altogether.
Spy suffers from a "situational" bias in 6s in that the only time anybody tries to play him is when it's patently obvious that the other team might be using a spy. So players acquire a wealth of experiences of spy plays not working out. Spy succeeds when it's *not* expected. Transition fights, and stop-start advantage pushes, i.e., when teams don't want to force uber due to having large advantage so they push slowly. It's doubly effective because in such situations players don't have time to look around, being focused on the opposition, nor are positions as established and stationary as they are during a last defense. The fact that spy is often a trade class, in that even if you do get a frag you'll die afterwords, is also not that punishing in such transitional situations, as teams often run suicide plays there anyways.
Engineer is a little different in that there are situations where he can be quite useful outside of last - again you *used* to see engineer brought out a lot more often, particularly when teams would push off of their own last, there was a *decent* chance that they would drag their gun out to 2nd in the event that the other team had uber advantage and was likely to push back into second. There is also the shutter door on Gullywash which Mackey used to use fairly regularly to conceal a sentry. Not to mention that using an engee *pushing* last is SUPER annoying and altogether viable.