Funny innit, you read a post and dont see it all.
The 'sharpish' unworn bit on the pad. Do you mean sharp (as a sharp object) or just unworn compared with the rest of the pad... You say you cant see how it would touch the sidewall...have you applied the brakes hard and looked ? Can it touch the sidewall...worse still force the sidewall in. That may allow the innertube to 'escape' where the tyres deformed under braking.
Even so....you still make no mention of tyre damage, i'd expect the tyre itself to be damaged if the pad could touch the tyre.
Real real close inspection of everything is whats needed.
Check the following..
Inner surfaces of the tyre for slits or thorns that may be still there from the first deflation.
Sidewalls of the tyres for damage from the blocks.
Bead of the tyre, make sure the bead is not showing through the rubber. (metal strands perhaps)
The wheel rim itself for splits or cracks.
Make sure your rim tape is seated properly and no debris or spoke nipples showing.