Could it be made to work? Maybe, but doubtful. Would NASA consider it? No, even if it would work.
The extra layer will add to the mass taken to orbit, which is a very precious commodity.
Whilst burning it off you would need to be able to ensure that the aerodynamic profile remained smooth and well understood. One of the big risks is that an imperfection in the skin disturbs the hypersonic flow further downstream, and results in a hot spot that can damage the craft. So a cold area that burns off slower may result on a hot spot in a critical area that breaches the tiles.
The big thing that means NASA would never consider such a plan is the need to validate the change. Since the craft has changed its aerodynamic shape for launch, all the launch data would be invalid. This especially affects the abort profiles, where the shuttle needs to fly back to earth in a hurry. Then validating and ensuring the safety of the craft is safe at all other parts of the mission, including emergencies.
Finally, any such coating would clearly be some sort of applied foam - not unlike the external tank’s foam. It would be subject to popcorning like the ET’s foam. At least it isn’t subject to the cryogenic temperature issues of the ET, but it would be subject to two weeks in space, so hard vacuum, and full bore solar radiation - which includes lots of UV radiation. Something most plastics are not fond of. I can imagine it degrading and popping like crazy. NASA would need to test and validate all aspects of this.
NASA have considered and rejected other changes to the thermal protection system. The tiles are far from a happy solution. But the cost, and time needed to change has never proven worthwhile. (Of course had they bitten the bullet 20 years ago - say after the Challenger accident) it would have been money well spent by now.) But it would probably costs many hundreds of millions and take two or more years to perform even a simple change to the shuttle. NASA have made some changes the the TPS, in less critial areas. The thermal blankets that cover the top of the craft have been modified over time.
Finally, remember the damage that doomed Columbia was not to a tile, but to the carbon-carbon leading edge of a wing. It is very unlikey that any layer of cushioning material would have helped here. After all it was a piece of lightweight foam that did the damage in the first place. Just putting a layer of foam on the wing would not spread the force, nor absorb much of the energy. The falling foam piece was totally shredded by the impact on the wing, but still imparted enough force to crack the RCC material.
The right answer would be to reimplement the TPS end to end with newer technology materials that don’t have the difficulties and maintenance headaches the current design has. But that isn’t going to happen.