So, let's assume, now, that you have bought all of this so far. What would happen after all of this if you were to reintroduce back into this approach the really big, balanced hands possibility? Or, some of them?
This gets interesting. Suppose Opener has the really big, balanced hand and can also bid 2H with THAT hand.
If Responder hears 2H and has a 5-card major, you will recall, he either bids 2NT (spades), 3C (hearts), or 3D (both). If Opener is able to have the really big, balanced hand for the 2H relay, and if he has at least 3-3 in the majors, then any of these calls will be easy for him to handle -- he supports the major.
Suppose that Responder bids 3H or 3S showing a stiff or void in the indicated major, 2-3 in the other major, and 4+ in each minor, with values? Again, if Opener has a hand with at least 3-3 in the majors, he should be well-placed to choose between 3NT and four of a minor, or 4NT.
Let's suppose Responder, instead, bids 2S waiting. Opener now bids 2NT, even with the really big balanced hands. If Responder has both four-card majors, he bids 3D. If Opener does not himself have a 4-card major, which only occurs with 3-3 in the majors and a really big balanced hand, Opener bids 3NT, and all seems well, again. If Opener has a four-card major (4-3), he bids his major and all is well. With 4-4 in the majors, Opener could perhaps bid 4C or 4D or 4H of whatever makes sense. Again, no problems.
Let's suppose Responder, instead, just bids 3C, asking Opener what he has. Opener can now bid 3NT to show the really big balanced hand. This creates one problem -- Responder could have one four-card major but not both, and Opener might have no 4-card major, one four-card major, or both four-card majors.
That's not great. But, at least we are getting somewhere, and on many sequences we are doing great. In this instance, sometimes we would play 3NT. But, Responder has the option of a 4C call as Stayman when he can afford the next level to check on the major. If Opener denies a major (bid 4D), Responder now can bid 4H or 4S as a minor flag, which would be nice, as well.
So, one might reintroduce some of the really strong balanced hands back into "Rexfordized Kokish," limited to maybe 24-25 HCP, with 3-4 in each major. With a 5-card major, 2-2 in the majors, or 4-2 in the majors, Opener would rebid 2NT directly (2C-P-2D-P-2NT), as with 26+ hands.
Of course, one might decide to include some other balanced hands that do not wualify into this structure. For instance, Opener might treat a hand with 4-2 in the majors as a major-MINOR canape with, say, 2425, 2452, 4225, or 4252. He might even treat 4243, 4234, 2434, or 2443 as a "major-MINOR canape" if that seems right and manageable. The auctions where this would create a problem would be when Responder shows a 5-card suit in the short major (bidding 2NT or 3C directly after Opener's 2H call), which seems to be not all that bad.