"I never gave much thought to the quality of the key oil or cork grease I use, I just assumed it was all the same.
What's the difference between a good and bad one? "
The big enemies of key oil are a gummy residue left behind after the more volatile components have gradually evaporated, and also moisture absorption which assists rusting of any steel. Typical engine oils usually contain a range of additives because they are especially formulated to carry out specific tasks in a pretty darn hostile environment of extreme pressure &/or temperature. They are often in places where there are no copper alloys to react with, so sulphur-containing additives are OK. Evaporation is seldom an issue. Residues might be filtered out. Detergents are often added to accommodate moisture, as a product of combustion, whereas in instrument pivots we want to exclude moisture.
Viscosity is important for each type of situation, eg a pivot point screw with higher pressure on a sax is very different from the low pressure, high surface-areas in the pivot tubes of delicate mechanism. WE want oil to stay put (in capillary gaps) rather than 'migrate" over a key to collect dust, which wicks the oil away from where it should be. We want the oil to have surface-clinging properties to protect steel parts from rust in a damp environment. When we "wick" the oil in form the cracks in the mechanism we often have to make compromises with the oil's viscosity, whereas the ideal is to remove each key to get the oil where it is needed.
As for olive oil! Jeez! Look on the pantry shelf at the sticky residue, like half-set varnish, that vegetable oils polymerise to, given time and especially with the encouragement of light and oxygen. Put that goop in a large-surface area pivot tube, and the spring will barely operate the key. It is also quite difficult to remove because it solvents work very poorly on it.