If you could "blow" through the ballcock it isn't the problem!
Take your hose back up to the tank and get it running through from there, block up the hose (in line tap?)to stop the water running down your legs (and emptying the tank) as you take the hose back to the toilet. Open your inline tap and you have water at the loo!
If it still runs too slowly, fit your pump!
Make sure your guests know how hard it was to get the thing to work.

Have fun!