Satsuki is it's own hobby. People grow satsuki. And then they grow some as bonsai and some as flower display trees.
The reason why it is hard to mix in other azaleas is because the other azaleas mostly flower weeks earlier. And therefore, they cannot be part of the same exhibition events.
Another reason is that at satsuki shows, in flower or out of flower, the trees must be of cultivar registered with the satsuki society. If not, the tree isn't technically allowed in.
I guess there could be non-satsuki azaleas or rhododendron entered in non-satsuki bonsai shows by people who just happen to do bonsai, not satsuki, and who just end up getting good material of this non-satsuki rhododendron.
I do not know how often that happens. Rhododendron yamadori usually are not good bonsai material no matter what the species, because they are shurbs, not trees. But landscape plants that end up getting pruned for decades may be much better material. When displayed in autumn at a bonsai show, it wouldn't matter if satsuki hobbyists thought it was a proper cultivar allowed in their hobby. I could see some bonsai artists in Japan, who don't do satsuki and have no real connection with that branch of the hobby, trolling a bit and displaying a non-satsuki rhododendron and breaking all of the norms that satsuki people like. Just not really aware of it. When I see kokufu videos, I see some satsuki, of common cultivars, as they would appear at satsuki shows. But I don't see other rhododendrons or azaleas.
Satsuki does not literally mean fifth month. The fifth month of the Japanese lunar calender is named after the rice seedlings they plant in that month, namely 早苗 or 'sanae'.
This was somehow bastardized into the kanji 皐 with the pronunciation just 'sa'. With 'tsuki' from 月 being the other character, for month or moon. The kanji 皐 also has its own meaning, namely swamp or shore.
This same month in the lunar calender also has an alternative name, which is 五月, pronounced 'gogatsu'. And this does literally mean 'fifth month'.
This is a bit like having a berry shrub that gives berries in september, and calling it 'September berry' and then saying it means '9th month berry' only to find out that 'september' literally means something else, namely 'seven'.