My opinions on pyro may be a bit old school since I grew up on the class "back in the day," but here it goes:
I watched you demo, and I saw what I thought I might see for someone having issues with stock: a kind of mix of degreaser and soldier gameplay. My first suggestion is this:
#in your pyro.cfg
unbind 2 #or whatever you use for shotty
unbind MOUSE2 #or whatever you use for airblast
Then switch back to stock axe. Play a couple of hours using JUST flame and the unstinguisher. It will frustrate the hell out of you, but it will mend what seems to be a mental disconnect for you with the class: flame is your main damage dealer. It can dish out 200-300 damage in under 2 seconds if your leading is true. Often in the demo I saw you give a token flame lick to someone, back off, then try to finish them off with shotty. To me this is the wrong mentality for stock (see my comment below on switch time). You wanna move in from the flank, circling around them (if possible) at the same time, so you make a Archimedean spiral of hell. You wanna get to the point where you flank a non-pocketed heavy and think "pig roast." Forgoing the airblast and shotty will also reinforce the flaking mentality.
Once you've got flame+circle-strafing in your mindset, then you can add in the airblast and shotty. I often saw you either airblast a weak class out of the range of your flamethrower (and thus out of your best damage), or airblast power classes right to a range/position where they are most effective against you. When airblasting people, use it to put then into the range that gives YOU the advantage. For example, pin soldiers to walls as you strafe away from their rockets, keeping the flame on them; don't airblast them to middling range where their rockets are still hard to predict but now don't do any splash damage to them. Think of it this way: just outside of flamethrower range, all you have is your shotgun, whereas the soldier has a shotgun AND his rockets. Use the airblast to assist your flame, deflect, harass around a corner, or get the hell away. For instance, there is typically little reason to airblast a solo heavy that you have caught off guard: better to use that time to lick him with extra fire as your run around him. If you get surprised around a corner by the same heavy looking right at you, set him on fire, blast him away, and get back around the corner to safety. Corners in general work really well with airblasting, as you can turn the corner, flame, blast, then get back to safety while they take afterburn.
As far as the shotty, use it to finish off light classes that moved out of range, though you should feel no shame in chasing down naked spies and engies with fire. If an enemy pyro is in flamethrower range, burn him instead. You generally want to handle solo demo/soldier/heavy with the flamethrower as well, since they are more effective than you at shotgun ranges. This means getting a good flank on them or being a boss with the reflects so you can close the range. Generally, only have the shotgun out as you travel wide open areas; the second you might come upon an enemy within flamethrower range, you should switch back to it.
I really feel you can't use the degreaser mindset with stock. The former only loses about 30dmg (for close direct flame) in the 0.23s weapon switch; the latter loses 102dmg in the 0.67s switch. Thus it's even questionable that switching to the axtinguisher with stock is much superior to just putting more flame on them, especially if another enemy is nearby. Fire, fire, fire.
If I had to sum it up, in the demo you remind me of the chemistry student in a clean white labcoat, goggles and gloves on, carefully opening a jar of solvent in a fume hood. You want to be the student setting his pencil on fire with the bunsen burner. Hope this helps!
