Based on previous years of adding things here, I've learned that if I want to find a linked conversation later, I might as well copy it to this blog. You can't count on the web topography.
Sorta like the landscape of the Black Zone.
Chaos-Based Campaign? theRPGsite
RPGPundit pretty much sums up my thoughts, but I've tested a number of assumptions in trying to mesh Corwin's story with Merlin's story.
First there is the cultural issue: the Chaos Lords are in competition with each other, sometimes to fatal levels. As we see with Oberon, the elder Lords do not give information and advantage to the younger scions of the Houses. They make them work for everything. It is logical and cultural to think they pare down the privilege of Logrus to the very few young people they either control or are sure are the best candidates.
Would you want a bunch of semi-insane hotheads running around testing their Logrus might? The canon tells us there are skulls littering the floors of the Logrus trials. As an Elder, how many of the best and brightest of Chaos would you kill to get the few survivors?
Second, given that we are "following the adventures" of gifted people, you could assume that all Chaos PCs have Logrus, but that's hardly the most interesting or effective combination, as others above have said.
Third, when I've played Chaos PCs, I've tried both Logrus and non-Logrus persona. I have to tell you the non-Logrus PCs have been more effective, if only because the other PCs were throwing a bunch of points into something that did not "niche" their character. IOW, when everyone has an ability, it diminishes the drama.
Fourth, yes, certainly there are other abilities. You can load up on Attributes in a way that makes you good/better than amberites. You can go for Exalted shaping. You can push Sorcery to places that would make you a legend in Chaos and Amber. You can explore demonology or spikards.
In short, if you work at it you can get a leg up on spotlight time with an offbeat Chaosi.