(disclaimer: the below in only my personal take on the above question and may not be a complete answer. )
I can't speak for other iris hybridizers, but in general, a decent bud/bloom count ensures a longer bloom period for that particular variety. It is not an absolute, but a MUCH desired trait in new cultivars. High bud counts and bloom potential and garden presence/show bench material seem important to a significant number of irisarians. The other traits you've mentioned are important/essential, but if you are planning on people actually paying for the privilege to grow your seedling, it all starts with the bloom impact.
(Again, IMHO, nothing more)
Cultivars with only a few, lets say, 4-6 buds per stalk would either have to throw a multitude of stalks for a good garden show, or the individual flowers would need to be very long lasting, since each bloom lasts only a few days.
Most tall bearded iris open their flowers successively, starting with the terminal bud and then working downward. Such an low bud count iris, if its hardy, grows well and blooms reliably, and increases well, could be introduced, but mostly likely would not win any awards, and the hybridizer would need to be honest about the trait, AND market it on its other merits, rather than multiplicity of bloom.
So to answer your question -- you don't absolutely HAVE to intro a seedling IMHO with a high bud count, but decades of iris shows, competitive show exhibition, regional and national award symposium competition, and every higher retail prices for new hybrids have set an artificial high bar for new iris. Plus the fact that there are over 80,000+ registered iris in the bearded class. One needs to carefully evaluate and consider if your candidate measures up.
There have been Dykes Medal and Wister Medal award winners that did not exhibit all of your garden criteria, Elsa, that became popular with the iris public. Decadence, and Sea Power are 2 that come to my mind.
However, I don't believe that great health and garden habits vs high bud counts and branching are mutually exclusive and I firmly hold to the notion that most relevant criteria can be achieved with all the breeding material out there. The main issue is educating hybridizers both new and experienced, thoroughly on what is now the accepted standard(s) for flowering and gardenability that the irisarian gardening public expects.
Apologies for my long winded answer, im sure there are gaps that need addressed