I wrote extensively on this in the old forum. Here's the short version, reinforced by my personally going to the place and interviewing one of the directors, buying histories outside, and reading up.
The original Butokukai was monitored by the Ministry of Education, and patronized by a (distant) member of the Imperial family, one of the 'cadet' branches (something like 18th in line for the Imperial throne if everyone else piled in a single bus and went over a cliff).
In 1942 or so (IIRC, not without my notes), the original Butokukai was disbanded and a new one established, new charter, etc. I found some of the unpublished original notes for the new charter tucked away in an obscure book, documenting the change to its objectives. The Chair was Prime Minister Tojo Hideki, vice chairs the Ministers of the Army, Navy, Education, Health, etc.
Postwar several of those folks were hauled off as Class A war criminals or otherwise banned from public service for life. The status of the Butokukai was sort of in limbo, there were a bunch of administrative and public maneuvers, etc. that went on for almost two years. I found and interviewed the US Army officer who actually negotiated what he thought was going to be a generous reform that would allow it to continue; much to his surprise, the remaining Butokukai reps announced they had no intent to reform, and shut the whole thing down. (The new Butokai story that it was forced to disband and the new one is simply their coming out from under that ban is simply not true. I have plenty of evidence to the contrary.)
The national sports / martial arts organizations quickly moved into the vacuum - the International Martial Arts Federation IMAF www.imaf.com,
patronized by Prince Lieutenant General / Uncle of Emperor Hirohito / Higashikuni as a capstone organizaton, the All Japan Judo Federation, the All Japan Kendo Federation, the Kobudo Shinkokai, etc...... All Tokyo based. All these organizations know they replaced the old Butokukai. In fact, I think all the kendo folks in the Butokukai get their ranks from the All Japan Kendo Federation - there is no room for splinters here, mostly, outside of koryu, karatedo, aikido, etc....
Later some gents in the Kansai area (Osaka / Kyoto), the original and traditional stronghold of the Butokukai, 're-established' the Butokukai, and laid claim to its heritage, etc. The Tokyo martial arts world apparently never noticed.
They did get the 'patronage' of one of the members of the 'cadet branch' of the Imperial family, Fushimi.... Higashifushimi something or another, the same family that had patronized the original Butokukai (it's late here, and I can't be bothered to look it up.... sorry if not quite right) who, like all the other members of the extended Imperial secondary prince lines became a commoner under the new Constitution, but they still call him 'Prince'. HIs family also actually provides the hereditary priests at a beautiful temple that houses the administrative offices of the new Butokukai. Beautiful place, I recommend a visit.
Its members seem to often claim some special connection to the Butokuden, but that is simply historic to the old organization and geography - the facility is owned by the city of Kyoto and available to any organization there that can apply to the lottery to use it. It was the site of the original Budo Senmon Gakko (Martial Arts Specialty School) which was run by the old Butokukai but all that changed postwar.
The organization is apparently first and foremost a Kyoto / Kansai area regional organization - mostly kendo, iai, and a sprinkling of other arts. It is simply almost unknown in Tokyo, even among kendo / iai types. I don't think I've ever met a Japanese judoka or jujutsuka that even heard of it before I ask, and I've asked a bunch. Most never heard of the old one, and those that have express surprise that someone co-opted its name. That was a long time ago, and while the originals had direct government sponsorship and oversight, this is not the case now (I think it's registered as an NPO with the Kyoto government, but that is a nothing burger, really....)
They also have trademarked the eight-sided00:49:42 mirror symbol of the old Butokukai, which I find highly dubious, as I have a couple dozens of them in various forms and the original organization is long long gone, long out of trademark time limits, so I reckon I could use my images as I wish. But I am not a lawyer....
Regarding promotions, I once saw the application paperwork for someone to join overseas, and it seemed to be a paperwork drill. I don't recall if they asked for a video or whatever. Probably not at the lower levels - part of that may be the difference between Westerners and Japanese - shodan is simply a serious student here, while overseas some think that's close to finished....
I think it's pretty harmless, and they've apparently had some decently attended international seminars. I would assume that most of the folks are decent - I've met a couple that were outstanding, very good martial artists and Nice Guys - but would not link the new, third generation organization to the legacy of either the first or second generation.
There was apparently a serious reorganization within the last year or so. Not sure what that portends.