Even if you don't look specifically at flakes, VA's in general (regardless of cut, but probably even by cut) are far more popular, and a lot of people are almost proud of not liking burley... to me, akin to that Portlandia skit or whatever (never saw the show just familiar with the clip) of a bunch of snooty people "DID YOU READ?" and "I don't even own a TV", though the proudness of not liking burley pales in comparison to the people tripping over themselves to proclaim their hatred of aromatics like only children and wimps smoke aros and real smokers only want unadulterated pure tobacco taste, because they're a real man (at least that's how it comes off to me). I mean someone saying they don't like aros when it's relevant isn't really the issue (and it's perfectly fine to not like them), it's more the people that go out of their way unnecessarily to proclaim their dislike of them when it's not really relevant (like if someone is specifically asking for aro recommendations and that's what the thread is about, it helps nobody for you to come in to say "I don't like aros").
Hell, another forum someone started a thread about what's your "final four" or something about what you're currently smoking between vas, aros, burley and English and the number of people putting down blends for 3 of the categories and putting "I don't smoke burley" for the burley category was like 1/4 of the responders on the first two pages. And there's others still that the only burley they'll smoke is ODF/Kentucky.
And then there's still occasionally threads that crop up from time to time about "I thought burley was just cheap floor scrapings for cigarette tobacco, boy was I wrong" or some such.
All that is to say, burley of any cut is not immensely popular compared to other blend types... at least as far as forums are concerned. Many think of burley as either "noobie/beginner" tobacco or "codger stuff". VAs are probably the most popular blends.
And that's a big part of why you don't see more burley flakes (not as much demand from consumers as for VAs, at least as it appears on forums and for boutique blend buyer types).
If there's more to the story than that, I don't know. I do know burley is a little more difficult to grow than VA (at least I think that's what I read at some point), which could mean less production compared to VA which could translate to less use, or less familiarity.
Then probably other factors like, too many are the same because they're all trying too hard to be an ABF clone, which is one of only a handful of burley blends that get regularly recommended, and not trying to experiment with burley (if you think burley is one-dimensional and something you can't explore very far as a basis for a blend, you've never had LJP's burley blends) or thinking it can't do much other than add nicotine to a blend or alter the ph of a VA or something also probably contributes
