Yes, and even chemical rockets make use of the exhaust as effectively coolant. But the problem is that then you have to carry enough mass to carry away heat. An

Yes, and even chemical rockets make use of the exhaust as effectively coolant. But the problem is that then you have to carry enough mass to carry away heat. And suddenly your rocket is 99% expendable coolant and only 1% payload (or less). Or to put it another way, nuclear-thermal rockets might have an Isp of 3000 or so, which is amazing compared to a chemical rocket which might have 1/10 of that ... but an ion engine can have an Isp of 30000-70000, which is what makes a 550-AU trip even remotely practical. I don't have the math right now to estimate how hot a conventional combustion chamber would have to be to get ion-engine-like exhaust velocities through a rocket nozzle/bell, but I think the answer is probably "ridiculously hot".