I don’t know whether the system you describe exists, and I don’t have time to deeply analyze whether I think it would be an overall good system. Sorry.
However, I think the problem you call out here (specifically with STAR) is not really a significant problem in the real world, and is simply the result of clustering the voters into two hard edged groups, with no fuzz around the edges.
I agree with you that B is the “best” result, while also arguing that STAR is probably doing the right thing by picking A. I know that sounds contradictory, but hear me out.
The thing is, the majority have A as a first choice. That means the median voter’s first choice is A. And I think there is good reason to say that – if it can be determined who the median voter is – the best system should always pick that voter’s first choice.
The reason for the seeming contradiction, as I allude to above, is that your example is very unnaturally contrived and hard-edged. A more realistic scenario might be this, which is almost identical to yours:
Red 49.5%: A B C D
Purply-Red 2%: A B C D
Purply-Blue 1%: A B C D
Blue 47.5%: A B C D
I’m a graphics guy, so I would describe what I did there as simply applying a small amount of blurring. Thanks for using Red and Blue as the voter groups, since blurring red and blue gives you some purple
In this case, the median voter is Purply-red, even though they only represent 2% of the total. And STAR should pick B, first choice of that median voter. All you need is a tiny percentage of people who actually like B the best (here 3%, both the Purply-Red and Purply-Blue groups), but it could be even lower).
Even more realistic would be to blur it some more, and have more like 10 to 20% have B as a first choice, but I went with a small amount of blurring to make it very close to your example. Regardless, I hope you can see that it is very, very unlikely to have absolutely no one rate the middle ground candidate as their top choice, given that everyone has a pretty high opinion of that same candidate.
There is also the oddness of having B be so highly rated by both sides. You’d think that if so many rates B as 4, there’d be at least some who would rate B as 1, 2 or 3. If you were to try to explain this scenario in a “Map of Tennessee” sort of thing, I don’t think you could do it no matter where you put the proposed capitals and the voters.
The more people you have voting, the less likely it is to have these highly polarized situations that seem to show a flaw, but they really just show a very unlikely (i.e. weirdly contrived to trigger a flaw) scenario.
Also, I don’t want to pick on Approval, but you can’t know for sure that it will handle this situation better, because it is so dependent on who the voters think will be the front runners. A bad poll, or incorrect speculation as to how voters will set their thresholds based on a poll, could cause any of the three to win (either in your scenario or my “blurred” one). For instance say the polling says that A and B will likely be the front runners. Now the savvy Red voters will approve only A, while the savvy Blue voters will approve both B and D. B will be at a big disadvantage, getting less votes than they would have, if the polls predicted it would come down to A and D.
And B would still be disadvantaged in my more realistic scenario (i.e. the blurred one with some purple voters), under Approval, if there was polling or speculation that lead people astray.
So STAR might be sensitive to highly unnatural clustering of preferences, but Approval can be sensitive to incorrect polling or speculation.