Update:

I removed the master cylinder and observed the O-ring was worn/expanded. I took a working master cylinder from a parts car and swapped them, and now the brakes seem to be a lot more responsive.

It still requires pushing down 1/4 of the way before brakes engage, but it responds quite rapidly once it starts resisting. The pedal no longer sinks as I hold it down. I could be wrong, but the O-ring would explain the pedal sinking (ie leaking air), as it's supposed to seal the air in the booster. I probably just could have replaced the O-ring, but I didn't have a spare one on hand and the spare master cylinder was already around.