I don't think US military adoption is necessarily all it is, but that is a factor. I think the BHP is troubled by being super generic looking to the vast majority of people out there and for some reason it just never gained any notoriety in general. Even though is was seriously advanced and miles ahead of its' time when initially designed and manufactured (pre-WW2) you barely see it featured prominently in films or games (especially compared to more popular pistols) and if you do its' looks will cause most people who only know a tiny bit about guns to mistake it for a 1911. It is odd because some of the most elite units of Nazi Germany used it as well as SF and Airborne elements from the commonwealth on the Allied side, then it was our standard issue pistol until the Glock came along in the mid-2010s so it did many decades of service. However it did become rather obsolescent in the 80s once the M9, P226 and similar started hitting the scene and obviously the entire internet and FPS games and airsoft didn't take off properly until the 90s. An awful lot of the guns that are cemented in to the popular parlance these days are designs that were all the rage in the late 80s-early 90s. The BHP sits in the awkward middle ground where its' WW2 service isn't much known or documented, yet its' popularity had faded come the golden age of action movies and the birth of our modern shooting games (virtual and real).
It also kinda suffers from being too good in a way as there's nothing that stands out about it, it's mostly a boringly effective handgun and even worse its' plain looking to most folks as I mentioned. It's not a big .45 with chunky rounds, doesn't have a crappy single stack mag capacity like the P-38, no weird action like the Luger, no crazy plastic-fantastic like the Glock (that blew people's minds in the 80s). Mediocre triggers and sights on most of them as well, so while it is pretty good it's not an amazing performer either and didn't have the best rep for reliability in some military circles. Never had any really large community/industry around it doing modifications either. There was a little bit when pistols were still legal and there was more competition in the UK/Europe, but obviously the 1911s and Berettas would have been 'the thing to have' for civilians in the US.