I get the same feeling from Isshiki.It's difficult to guess what Isshiki is thinking. He's one of those characters I expect to someday become the center of attention somehow.
Well obviously there will be other schools. It's difficult to imagine that a country like Japan would only have one, but on the other hand it's difficult to imagine that there would be that many. In the end Japan is densely populated, but not that big of a country in terms of land. This is simple part of the aura that Tootsuki is supposed to have. In reality a school like Tootsuki is not realistic. It does need to be said though that Tootsuki is not just for anyone. In fact this chapter has solved a question I've been pondering on for a while now: how did Souma get in so easily? How did he pay for it? It has been emphasized quite a lot that Souma sticks out like a sore thumb. But now that it has been revealed that Senzaemon requested that Souma would be enrolled, it has been pretty much explained.
And its funny that you bring up the issue of how Souma could possibly afford to go to school there; I actually never wondered how he could afford, though I guess I know now.
Honestly, I thought that Azami simply wouldn't acknowledge the existence of the PSD. I thought that he would treat it like some dinky, run-down place in the badlands, and that his hubris might prevent him from seeing the members of the PSD as a threat to his plans. That's the feeling I got from him when he walked off the stage after his speech and the Sendawara twins said that he didn't need to feign innocence because he was confident everything would go his way. I figured his attitude would be more like: "try though they might, there's nothing they can do to stop me. Their time would be better spent making the most of these coming days, because before long, they'll all be gone ..." or something like that. As if he would eventually be rid of them, I figured he had 'more important matters to tend to' and might not make any moves against them specifically.And that's the exact reason as why. Hisako has been with Erina since they were children and she just started opening more to both Erina and Souma. So much even that Erina was surprised by her attitude towards Soma. And on that exact moment she gets fired. What's she supposed to do then?
That Ishiki is part of the PSD, is I think not that important. It has been said before in the series, but PSD enjoys within Tootsuki quite a lot of independency to the point it's practically an independent nation. That alone seems to be enough reason for Azami to not tolerate it as he's striving to get the power in Tootsuki centralized. If that same PSD then would become the core of the resistance against him, which is very plausible, it only gives him more reasons to get rid of it. So that Ishiki is part of it I don't really think would be that much of a concern. Ibusaki said it this chapter, that he was wondering why Ishiki was left out from all of this, which means Azami already has somewhat separated Ishiki from the other Elite 10 and deemed him to be questionable.
But like you said, PSD is almost like an independent nation in itself. It's quite a distance away from the campus and appears desolate on the outside. A lot about it gives off he impression that it doesn't belong. When I think about it, I suppose the exact opposite is possible as well, that because the PSD "doesn't belong," he'd be all the more eager to do away with it. Perhaps one of the reasons he could be so confident is because he had already taken preemptive measures against places like the PSD. Now that I read through this thread again, he seems like more of a shrewd man than I initially made him out to be, especially considering the way he separated Hisako and Erina and the way he apparently instilled fear into Erina from a young age.
Seems I may have underestimated him .-.