I've certainly seen Burnelli's designs before, when I've searched for aircraft using the lifting body principle in any of its forms.
Reading various pro-Burnelli links it would seem that the entire worlds various aeronautical establishments all managed to miss a very good sort of boat.
Even if I dug up very detailed aerodynamic analyses of this or that Burnelli design compared with an OTL-dominant alternative--say, I got ahold of the Army Air Force's report on the CBY-3, versus the characteristics of a DC-3, I don't think I have the aerodynamic understanding to judge the validity of the reports, nor to weigh the political dimension that might be influencing these allegedly objective papers.
However, consider the number of opportunities the Burnelli designs (if not Burnelli himself!) had to be adopted if indeed their advocate's claims of their superiority are even half-true.
First, the various competing airframe makers of the USA. Cartel or no cartel, they certainly did compete with one another, sometimes driving one or another out of business (to the point that today there are effectively just two major aircraft firms in the USA, only one of which makes large passenger planes). Surely if one could reverse a tailspin of falling market share they'd revive, or claim to reinvent, such an integrated design, if it would save them--even if they had to belatedly or posthumously settle with Burnelli and rehabilitate his reputation!
Second, the US aero industry is hardly alone even in the capitalist West. Britain and France, and eventually other European nations, increasingly in the forms of multinational partnerships, all rival the major American firms. They too would certainly want to grasp at any straw to pull ahead of such American behemoths as Douglas (in the prop-plane era) or Boeing (in the jet era).
And of course at least until the late 1980s, there was the Soviet bloc to consider, and even to this day the aeronautical enterprises of the PRC (which would not have been much of a factor in Burnelli's lifetime to be sure!) Particularly in the years after Stalin's death, one would think that if a Burnelli type design had the sorts of commanding merits his champions now claim, some designer's bureau somewhere in the Eastern Bloc would have gone right ahead, with stolen blueprints or simply arriving at similar conclusions from similar premises.
After all, Burnelli's concepts were applied to a broad range of aircraft--piston and jet engined fighters, bombers, and heavy as well as light transports, in both high and low subsonic speed ranges. If one of them were in fact strongly superior in any of these ranges I'd think someone somewhere in the world at some time would have built them in some numbers and they would become a standard. If not in the allegedly cartel-ruled West, then surely in the rival East! And having once proved the concept viable in one niche, all the others would get due reconsideration.
I'm not absolutely sure that Burnelli could not possibly be the victim of a short-sighted and narrow-minded conspiracy, and that sheer stupidity and laziness does not account for the rest of the world's failure to attend to his suggestions. But it does seem unlikely to me.
That said I don't want to jump to any conclusions as to just why these designs would not be attractive. To be sure, making a wing thicker does tend to increase the drag--but that's relative to its length, as is its thickness ratio. It doesn't look to me like Burnelli's length to thickness ratios for his fuselage sections are all that much thicker than wings typically had in the subsonic era. Bearing in mind we are after all completely eliminating the traditional non-lifting fuselage and adding in considerable lifting area, which in turn allows the dedicated wing structures to be smaller in area thus eliminating another major section of the drag area, I'd be surprised if in fact the overall drag came out as any higher than a conventional plane of the same weight.
I believe the Burnelli boosters also claim the unconventional fuselage was not actually very difficult to make, compared to the standard box or tube shapes. If the opposite were true that might explain why airframe makers were so reluctant to deal with it, but I can't imagine that they'd be so terribly more expensive to make.
I wondered how off-putting passengers would find the layout; in particular it would seem there would be fewer window seats in proportion. But that would certainly be irrelevant for a cargo plane. (Windows themselves are a structural weak point and it might seem silly to worry about providing them, but generally speaking sites to put windows on would also be sites to put emergency escapes at, and these are vital safety features that are mandated by the regulations, so a design with fewer opportunities for windows is also one that is trickier to supply adequate safety exits for).
All in all, it is unclear to me just where the designs fail. But it seems evident, from the verdict of numerous rival aeronautical establishments which have all given the option a pass, that there is something wrong somewhere.