the thing is with CRT, the parts that are hot and using most of the power is not silicon, when you talk chips, diodes and transistors yes 70W is well in the realm of needing beefy cooling, when you are talking about incandescent light bulbs, cooling it would only reduce its brightness, and electron guns are an incandescent light bulb with a 3rd high potential contact, the cathode need to be red hot for the electron to be able to "jump" and form the "cathode ray". although if you only put a fan in with a dust filter you are unlikely to cause any arm, or even any difference, without the dust filter it could be really bad though as introducing more dust can cause shorts and overheating of the flyback.
one thing you could do if you trust yourself not to touch the high and mains voltages inside is to run it open and look at the boards for component getting hot, other than tubes/valves, i would not expect many to do so though, just so that you do not waste your time and money on modding CRTs