As for turbine engines, it doesn't actually rely on air cooling, but oil. But can I say that turbine engine doesn't need "exterior" surface cooling like a piston engine cylinder does?
Thats fairly correct, although oil cooling isn't a factor in cooling the exterior of the combustion chamber(s). As already discussed is "cooled" by compressor discharge air.
In fact approx 75% of air flow is for cooling, the remaining 25% for combustion.
BH