The issue is that the inspire minions just aren't that great. Not enough to form the basis of a deck, at least. There's maybe one or two good ones like Savage Combatant, but most of them just don't work. Considering the fact that such a deck would want/need to hero power on just about every turn, you can't rely on overcosted minions across the board. They're nearly all horrible the turn you play them, and if you have to hero power every turn, you're perpetually behind on mana.

I see no way for such a deck to be playable. You'll drop a 3/3 on turn 3, then on turn 4 you'll hero power for some mediocre effect like gaining a dude, and then you can play a 2-drop. Now what? Are you gonna play another 3-drop on turn 5 and hero power again? How can such a deck have a snowball's chance in hell of keeping up? When are you supposed to even get enough inspire minions on the board to start snowballing?

If you play these minions on curve without hero powering, they just die because they all suck ass by themselves. Each and every single one, without exception, has below-par stats. You cannot keep them if their inspire effects don't compensate for their shortcomings. But if you're hero powering every turn, you're perpetually two mana behind curve and none of the inspire effects are strong enough to compensate for this unless your opponent generously leaves your board alone for numerous consecutive turns. It's just unrealistic.

The hero power support minions won't do the trick, either. So you play a Fencing Coach on turn three so you can play, I dunno, Spawn of Shadows on turn 4 and hero power afterwards. But the 2/2 died on the opponent's turn, guaranteed. How is this supposed to work? It just doesn't compute. None of these things have inspire effects that actually match the disadvantage of playing weak minions and having to spend mana every turn maintaining them.