Install new master cylinder without bleeding?

Sounds like you figured it out, congratulations!
The reason you had so much trouble bleeding the whole system before was because you hadn't bled the master cylinder first!
And my current method doesn't require a helper, just a piece of clear hose, and a bottle you can suspend ABOVE the bleeder screws. Open the bleeder, get back behind the wheel and slowly pump the brake pedal.
The air bubbles want to rise in the tube, keep pumping till bubbles stop, close bleeder, go to next bleeder.
Don't let master run out of fluid, or you'll have to start over.