Yes, it should work on most surfaces.
My preferred primer is Zinnser shellac based BIN primer (not 123), two coats of that and you should have an excellent base on which to apply your two coats of an oil based eggshell paint.
I reckon any decent painter should be able to paint wooden doors... if the paint is chipping off a repainted kitchen then it hasn't been done correctly.. if it hasn't been done correctly why assume the doors were the problem and not the painter?
Painting isn't rocket science... if there are no contaminants on the doors (i.e oil, grease, dirt etc) and you use a specialist primer designed for the purpose and a high quality top coat I can't see what the problem would be... (although there are good and bad techniques, and wrong ways to paint a door... center panel done first vertically, then horizontal 'rails', finally vertical 'stiles'... when using oil based paints you can bate on the paint and then do a final light brush over in the order I've described to leave the correct brush marks, not possible with the BIN primer as it dries too quickly, don't rebrush it)
It is of course good advice to check back on previously painted doors at intervals.. i.e after second door, check first one for runs... after third door check previous two etc etc... it only takes a few seconds to look them over and it is time well spent if you can rebrush out a run before it dries...
Tesco are selling very cheap brushes which work well and don't lose bristles. Only 2 euro for a 3 inch brush which is fantastic value.