I agree with all those concerns. I don't know if the complexity, gimbaling, and cooling considerations can be adequately solved.
But this idea of a simple skirt around the rocket engines seems to have a lot of merit compared to side mounted ramjet boosters. It should improve efficiency across all altitudes and airspeeds. The P51 Mustang channeled bypass air around its exhaust pipe and got a noticeable increase in thrust at 500mph. The drag created by air intakes must be contrasted with the drag created where a cylindrical fuel tank ends a meter or more before the exhaust plume begins with a conventional rocket. Combustion is difficult to achieve at high altitudes and airspeeds, but heating bypass air works basically the same way. Fuel may be cheap, but good fuel tanks are not. The question is how much a skirt would cost in terms of weight and money compared to the fuel tanks you would need without them.
If a relatively thin titanium alloy skirt could withstand the temperatures, without active cooling, it might be a better deal.