As i am not an expert, and my English are not very good i will suggest you check
1. The bearings
2. Try to blow or suck from the inlet or outlet tube of the cooling system. If no air can pass through try to clean it because that means that even the air cant pass trough the cooling system in order to spread heat of the spindle to the environment
3. One basic question (because you don’t mention it) is if you use the spindle to cut something or you just turn it on with no cutting load. If you use it to cut any material check the bit condition
4. If you use a VFD (inverter) in order to control spindle speed, try to check it. May be bad electrical parameters cause overheating

I can think of something else

Vagelis