I find your thought process interesting, and I like how you're trying to think of new angles. But I think there is a slight flaw in your basic assumptions. Specifically, whatever information those three cards can tell you about the hole card, they will also tell you about the next card. Suppose you can correctly modify your count information using knowledge of those three cards and you calculate correctly that the probability of the hole card being a 6 is now X rather than Y. If so, you will also know that the probability of the next card being a 6 is also X rather than Y. Optimal strategy would account for the shifting probabilities of all unseen cards, not just the hole card. This is exactly the same sort of information that a count system provides. That is, it tells you about the composition of remaining cards but nothing about their order. The better the count system, the better (on average) your decisions will reflect the actual composition of the remaining cards. If you could correctly modify your count information in the light of the 3 cards you are considering, you would have a slightly better count system than you would otherwise have. In a sense, you have taken a (very small) step towards precise knowledge of how many of each card remain, and what this implies for strategy. Correct modification of your count that factored in all the cards in the present round would be a larger step. Keeping separate count of all thirteen denominations from the start of the shoe and calculating the optimal strategy on the basis of the remaining densities would be the ultimate level for this type of strategy (which monitors composition but disregards order).
To emphasise the distinction, compare two basic scenarios. In the first, you think you spot enough of the dealer's hole card to know that it is card A, B, or C with equiprobability, but not card D, E, ..., M. Here you have much stronger information about the dealer's hole card than you do about the next card. You know the hole card is either A, B, or C. In contrast, the only additional information you have about the next card is that it will be drawn from all unseen cards, where these unseen cards have been depleted by either one A, one B, or one C. That is, there has been a very slight shift in the probabilities relating to the next card (the shift being smaller earlier in the shoe than later), but a dramatic shift in the probabilities relating to the dealer's hole card. In this case, the information about the hole card is of overwhelming importance, and will have a much bigger impact on optimal strategy than the weak information about the next card.
Now consider a second, very hypothetical scenario. The dealer accidentally deals herself two hole cards. You spot one of them well enough to know that it is either card A, B, or C. The dealer notices her error and calls over the pit boss. The pit boss picks up the two cards, concealing their identity, and walks away to make a phone call. After a minute or so, the pit boss comes back and announces that one of the cards will be the dealer's hole card, while the other will be the next card. (The PB says that they don't like to burn cards in their casino because, for given penetration, dealer speed, player speed, and number of chip purchases and colour ups per hour, it causes shuffling to occur too frequently, thereby lowering rounds per hour and the casino's hourly return. The PB uses this exact explanation. ;-)) As a result of the PB's decision, you now have information that pertains equally to the dealer's hole card and the next card. It is now more likely that either the hole card is A, B, or C, or that the next card is A, B, or C. In this case, to play the hand optimally, you would need to factor in both effects. There is a symmetry to the information in this second case that is absent in the earlier scenario. This symmetry is also present with the approach you are suggesting.
