Bill, As I posted already, my thinking is that you are correct that this would not be a useful design.
But, that is just the gut feeling of us two who have calculated such things for other ideas, not this one. Wisetoe asked for the simple calculation that you performed for him. My comment is simply that the calculation asked and given is not really what is needed to address the proposal's effectiveness, because it does not include the beneficial aspects I delineated. But I do NOT think those aspects can overcome the 200:1 weight difference. Here is why:
Based on
https://www.nas.nasa.gov/pubs/stories/2022/feature_LAVA_Parachutes.html , I'll guess that a parachute could be deployed at about 1,500 mph = 670 m/sec. So, the total velocity change still would need to be about 10,000 m/sec. To accomplish that ONLY with steam emitted from the cooling jacked, and limiting the total amount of steam to the weight of the current ablative heat shield, about 4000 lbs = 1,800 kg (see
https://www.space.com/22036-nasa-orion-spacecraft-heat-shield-photos.html), that steam would need to be emitted out forward facing nozzles at something like (10,000 m/sec x 8,500 kg / 1,800 kg =) 47,000 m/sec. But, the exhaust velocity of the Hydrogen/Lox powered J2 rocket engines (which emit superheated steam) is only about 1/10 of that. See
https://en.wikipedia.org/wiki/Rocketdyne_J-2 .
Worse, that J2 performance is when the gas is going aft, not forward, and the back pressure on the nozzle is nearly the vacuum of space. The chamber pressure of the J2 engine is only 763 psi. The effective back pressure on a nozzle facing forward is dependent on its velocity into the air and the density of the air. I did a quick 'n' dirty scaling of some numbers I won't try to explain here, and it looks like the "max Q" for the Orion capsule reentry could get close to the J2 motor chamber pressure. So, the thrust would be very small when the heating was very large, unless the temperatures and pressures inside the cooling jacket far exceed what we can handle in a H2/Lox rocket motor.
Further, the surface temperature of the reentry craft would reach about 5,000 degrees F if it is not cooled, but the metals that could be used would need to be kept to more like 2,000 degrees or less. So, if it gets to the point that there is not much retro thrust, the cooling is rate is going to be very small until the pressure and temperature inside the cooling jacket get very high. So, now you are talking about making a rocket motor with a diameter the size of Orion capsule diameter out of super alloys, and make it far stronger than the J2 rocket motor. That in itself would have to be very heavy.
Exactly how all of this would play out in terms of design requirements would require a lot of internal and external thermodynamic and aerodynamic calculations integrated over the entire descent path. Obviously, I am not going to try to do that myself. But, I think the issues I raise here should be sufficient to convince folks that this is not likely to be a feasible substitute for a heat shield.