Brake conundrum

Did you bleed the brakes with the engine off? Engine can't be running when you bleed the system.
If that's good, then it's a bad booster. If your bad master cylinder leaked ANY fluid into the booster, it's gonna go bad. And the symptoms are similar to what you describe.
If you don't believe me, then listen to Booster Steve. Around 4:50, but the whole video is worth watching.