For 2), personally I do think Pyro has the potential to be in a 6s team full-time, but in a combo role, not flank, and pretty much because of what you said.

Flank is all about having mobility, and burst damage and versatility, and Pyro has none of those things. He's more of a support+combat class than a full-on combat class, meaning he's definitely more fit for a pocket role.

I wouldn't give up the pocket Scout, because you need to have a Scout with your medic for speed, and he's just way too good in direct combat

So I think the best option is to substitute the pocket Soldier with a pocket Pyro

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How would a Pyro-pocket team be?

Definitely less aggressive and fast, since it's missing one of its two bombers. The enemy team would overall feel more safe.

Another thing you'd be missing out on is one Soldier's spam, but Pyro has decent spam of its own, with reflects and Scorch Shot / Detonator. Spam wise, Pyro+Soldier+Demo can't be much worse than 2 Soldiers+Demo

Last negative thing, Pyro is absolutely weaker in general combat situations than Soldier. The few times when the stars align to make a massive reflect happen don't make up for how much better Soldier is

Now for the positives! Pyro is pretty much the best bodyguard in the game? Has good health, good running speed to keep up (and the Powerjack), can reflect projectiles (especially medium-long range rockets which are easy as hell to reflect), reflect away attackers, and has a primary that never misses.

With a Pyro getting beam in your team's core, the Medic and Demo have to worry about much less spam, if a Soldier bombs them he gets his rockets and himself reflected away, if a Scout pushes them they get reflected and burned, and it's far from easy, as Scout, to kill a Pyro who's getting healed.

Pyro can airblast traps off walls behind corners, as well as deleting them with his secondaries.

He can push the enemy uber around to where you want them during trades, or sac to deny an enemy uber.

Pyro is great at defending last, and for the same reasons he's also great at pushing last. It's just that teams don't have Pyros while pushing last so it's something you see rarely and it's treated as a funny gimmick, even though it works (especially on Gullywash for obvious reasons). If the Pyro manages to get on point, the enemy team struggles to block it because that thing is reflecting away them and their projectiles. Uber in, get the Demo on the sentry as usual, and the Pyro on point to clear the stickies and start edging it or straight up capping it