PM5K, Many of the guys here who are advising you are pros that hang at PaintTalk as well. Meaning, you'll get basically the same advice there. There are no breakthroughs in painting that are going to solve your problem. You have a filthy situation there that requires some heavy duty stuff. And, I'm sure that's only one snapshot of a much wider picture. Do you really want to get down on your hands and knees and clean that mess? To me, rubber gloves wouldn't be enough, I'd wear a wetsuit. It's a rental, painting is an expense, treat it as such. What you like and what you prefer are reserved for PM5K the homeowner, and what works best in the quickest fashion with the least amount of expense is for PM5K the landlord/businessman. I suggested using oil based Cover Stain Primer Sealer, tinted to match your finish, and apply one finish over top, an off white for max hide. I suggest oil for two reasons, one is that lingering among all that filth is probably a lot of incidental stains that may bleed through latex alone--hand grease, crayon, food stuffs, water stains, etc. The second is to lock down all that filth. No paint sticks to dirt, but oil will stick better than latex. Labor is your biggest cost, use material to your advantage.
Choose a color
Have the cover stain tinted to match that color, use a 3/8 cover
Apply one top coat, eggshell, flat, whatever, use a 1/2 cover
A base coat of tinted cover stain will do all the work and you're free to use whatever brand finish you like. Two coats you're done, no cleaning, no spot priming stains and touching up, max coverage, and you'll have a much sounder surface, IMO, than with cleaning and two coats of flat. You went to PaintTalk looking for pro recommendations, there you go. No screwing around.