Question about Brakes / Master Cylinder

96 Chevrolet C1500 PU 2 wheel drive power brakes, front disc, rear
drum.

Hi All!

My truck evidently had a caliper get stuck and it quickly wore down the
inner brake pad on one wheel. The master cylinder fluid was down to the
bottom of the reservoir, so I wondered if I might have gotten some air
into it. Anyway, I have replaced the front calipers and pads, and used
the gravity method to bleed all four wheels. I inspected the back shoes
and wheel cylinders and they are OK. Now I have good brakes again, but
the pedal travels half the distance to the floor before the brakes
begin to kick in. I think it should have about an inch of travel :-).

So, my question is, if air DID get into the master cylinder, would it
have travelled to the wheels and now be eliminated by the wheel
bleeding, or should I actually bleed it now as part of my brake job? I
checked the master cylinder and it is dry (not externally leaking), it
doesn't bleed down under pressure, and the system is not losing fluid,
so I think the MC is OK.

Thanks,
Libby