once those had demonstrated reliability and safety
I'm under the impression from the idea's historical reception that neither the Air Force nor NASA thought there was a practical likelihood that that could be achieved.
Astronauts of course accept major risks, and understand that emergency systems may be very dangerous and still better than the alternative. Apparently both the Air Force and NASA thought this system couldn't be effective enough to be better than the alternative.
That's how heat shields work.
But no one...in the 1960s, and even today...has ever demonstrated a foldable, perhaps-flexible heat shield that can do what would be needed. The buffeting forces on reentry heat shields are prodigious; this one would have fold-seams, because none of the 1960s candidate thermal materials were flexible. All of the structure would be plastics with thermal-softening behavior. Pyrocerams and carbon-carbons are excellent insulators for separating plasma temperatures from high-temperature-metal structures, and could be faced with (pre-PICA-and-SIRTA) ablators, but how would you instead support such a system using only flexible plastics?
proper location of the center of gravity
Very hard to achieve when the device is to be shaped by the expansion of reactive polymer foams around the suit of an astronaut who will do his or her best to "assume the specified position", but likely will be in a hurry and perhaps will be injured.
A starting design technique for making an aero vehicle stable in the reentry environment is to make it longer than it is wide and tall. This object doesn't have that option, because it must store in so small a volume.
An alternate option is control surfaces. This object doesn't have that option either.
And, rotation and tumbling must be avoided. Tumbling in the reentry environment generates human-unsurvivable forces.