At the risk of belaboring this thread, I do think I've spotted the root of this disagreement, and can perhaps resolve it.
First off, in the case of Sport Pilot instruction received from a Subpart K CFI counting toward PP, I think we all agree that it doesn't. I don't like it, but unless or until the FAA acts favorably on our petition, that's the law of the land. Case closed.
But, we're talking here about Subpart H CFIs. Those folks are commercial pilots, with instrument ratings, who at some point held a Second Class medical certificate or above. So, the point of contention is: does a lapsed medical automatically turn a Subpart H into a Subpart K CFI? Well, the driver's license medical FAR certainly seems to say so, and I imagine that was the basis for the EAA Expert's response, cited previously.
However, there is a more fundamental question, which was addressed by FAA Legal about a decade and a half before the SP rules were promulgated: does the validity of a CFI certificate (under Subpart H) require a medical certificate? And, to that question, FAA Legal answered with a resounding no. A certain type of medical is required to exercise certain privileges of one's pilot certificate. Clearly, before LSA (and still true in a non-LSA), one must have a medical to act as PIC. But FAA allows a CFI to be a CFI, even in a certified aircraft, as long as he or she is not PIC. Examples include such a CFI giving FAA Wings flight currency training, and performing biennials (flight reviews) for appropriately rated pilots whose currency has not lapsed.
The medical certificate is attached to pilot privileges, not flight instructor privileges. On the back of my CFI certificate, it says "valid only if accompanied by Commercial Pilot Certificate #xxxxxxx." Nowhere does it say "and a valid medical certificate." Can I give you a flight review in your Bonanza, with a lapsed medical? Absolutely, as long as your two years haven't yet passed. And, I do that all the time. I did not cease to be a Subpart H CFI when I chose to let my medical lapse, I just ceased to be a legal PIC in your Bonanza. (BTW, this rule even applies if the Subpart H CFI's medical has been revoked or withdrawn! No medical required to exercise CFI privileges; it's a pilot requirement, not an instructor requirement.)
Now, how about training a SP to upgrade to PP? Same rule applies. I must have a valid Subpart H CFI certificate, of course, must have renewed it properly within the previous 24 calendar months, and all that. Nothing there about a medical; I'm still a CFI under FAA rules that predate SP. In fact, if the student is a current SP, and we're training in an LSA, and we're not flying at night, I don't even have to be PIC (though in an LSA I clearly could be). And, it could be argued, even if my medical were revoked or withdrawn, I could still give PP instruction to a SP acting as PIC, as long as my CFI is current, you're PIC, you're not under the hood, and we're flying in the daytime.
But, that was not the original question. As I read it, the question was: do SP instructional hours received from a Subpart H CFI with a driver's license medical, in an LSA, count toward PP training hours requirements? If we agree that (per the above) hours trained with such an CFI after the SP has earned his or her ticket do count, than how can it possibly be that hours legally trained with that same CFI before the student became a SP do not?
Yes, when you get ready for his or her night training, the prospective PP has to fly with somebody who has a current medical. Otherwise, all the additional training can be done (in an LSA) with a Subpart H CFI whose medical has lapsed.