'68 Auto 2+2: requires a double pump on the brake pedal to build any brake pressure. The pedal on the 1st pump, if not bottomed out, will provide feedback pressure on the pedal in the last 1/3 of the pedal swing and apply barely enough brake force to start slowing the car. Lifting the foot, and pressing on the brake pedal again (instantly) has almost instant feedback on the pedal and seemingly good brake pressure, enough to nose dive if needed. At a stop light, sitting on that second brake pump, the pedal does not go down and the car will not creep forward if there's pressure applied on the brake pedal. After a couple of seconds, if there's no pressure on the brake, the situation repeats and I have to double pump again to stop the car at a light/stop sign. I have bled the brakes twice, once by pumping the pedal and closing the bleed nipple before releasing the pedal for up-stroke, and the second time with a syringe, trying to suck brake fluid out. There's seemingly no difference to the original double-pump issue. Any suggestions would certainly be appreciated.

First thought based on experience is master cylinder.
The cups are wore out. Second pump flares them. Holding pressure with pedal hard down just tells you fluid is not leaking out anywhere else. Lift the pedal a bit..oh, usually an inch, to relieve pressure holding the cups out to the bore..wait 10 seconds or so, then press down. If it drops, rebuild or replace the master.

Also sounds like you may have some air in the system, a good bleed of the brakes might help

I agree with Dave C, First suspect is master cylinder seals, second suspect Servo Slave Cylinder seals are starting to wear