I have an air purifier / ionizer. It has little metal needles that apparently give off electrons when the ionizer is turned on. Is this effect the same as the electron gun from a cathode ray tube?
Your ionizer gives off ions because of the high voltage. Usually wires rather than needles.
The cathode ray gun emits a stream of electrons from the hot thorium impregnated filament which travel to the screen to activate the phosphors.
The high voltage in a TV set can ionize a bit of air in the vicinity of the transmformer if it is not well protected. Not a basic function.
Actually, corona discharge like that would be considered a malfunction to be repaired by the liberal application of insulating goop called “corona dope.” With the big CRT-based TVs, it wasn’t an uncommon problem - usually the most obvious symptom would be that the TV was hissing or you could smell the ozone.
So, no, a properly functioning TV isn’t very useful as an air “ionizer.”