I can't recall any Bloodname House's categorization change. But there is a tremendous amount of nuance out there.
We do see some rather prestigious Bloodname Houses have specialized. Nagasawa, Kardaan, are among the founders of their respective Clans but have chosen to focus on producing pilots, for example.
But overall it would appear that some rather prestigious Bloodname Houses have been given the General categorization.
Couple reasons why that must be considered:
1. Some Bloodname Houses seem to strive to produce successful warriors of all phenotypes.
2. Some Bloodname Houses seem to have been forced, or have chosen, to focus primarily on a single phenotype.
A lot of this has to do with planning by successive generations of Bloodname House Leaders. As well as examining the codexes of a particular Bloodline warrior's and evaluating what they are good at.
I love the description that accompanies the Faraday bloodname in FM: CC.
It's something like "Although Faraday is considered a strong General bloodline, it's pilots don't far as well compared to others."
So even within a General bloodline, there can be nuance. They can be really strong in certain areas and struggling in others.
I can't remember where I saw it, but I also once saw a pilot of the Kerensky bloodline, and it was noted that this was somewhat rare. Like Kerensky was being used across all phenotypes, but pilots were in the minority.
I think maintaining a General bloodline is just hard to do.
I'll compare this to countries who produce excellent Olympic athletes across many different categories and events. I'm talking about a country that dominates in both winter and summer Olympics and many different types of events. A country that is very successful in producing athletes that perform well. That isn't easy. Many participating countries chose to focus on a few areas where they have had outstanding results. They've just found the secret sauce to producing athletes who are great at <insert athlete event>. In categories and events where they haven't had any success, they cut funding and support.
So you need to have a solid array of warriors of every phenotype in the pipeline, both performing well currently, in the past, and among the sibkos. I suspect maintaining that is quite difficult. You'd also probably be best served by having a high bloodcount of 20-25 or so. If your Bloodname House is down to a Bloodcount of 10 or less, it could be very difficult to actively maintain branches of the proverbial family tree that cover all the phenotypes with successful warriors, basically every generation.
I highly suspect it's also not entirely up to the Bloodname House. The Bloodname House Leader has to work with the Clan/Khan/Clan Council who is trying to decide what sibkos to authorize. Of what bloodlines, of what phenotypes etc. The Clan in turn might require some convincing to let the Bloodname House Leader produce trueborns of a particular phenotype in which the Bloodname House isn't particularly well-known for. There's components of meeting the needs of the Clan here, as well as just politics.
EDIT: With that in mind, it's important to also note that there is likely a limit to how many trueborns a Clan will birth. Each year, and of what phenotype and bloodline. If Bloodname House A wants to expand and produce more, that "more" probably comes at the cost of a different Bloodname House who in turn will end up producing "less". This competitive component has never really been emphasized in canon sources (besides the Bloodcount), but it must surely be present. That means for a Clan to agree to produce more trueborns of your Bloodname House, or of a particular phenotype. You probably need to have some pretty compelling evidence that this is the path the Clan should take.