I dunno, there might be several reasons.
--Senzaemon was getting old. He could pass on, and either there would be no new director, or Erina would graduate Totsuki right into that position.
--Those of the Elite 10 who voted Yes to Central would be afforded far more power than they would have if the current director stayed. They could graduate Totsuki as billionaires or be world famous.
--The new director somehow has dirt on the Elite 10 and their families (Totsuki generally has wealthy students enrolled) and they have been forced to vote Yes.
Senzaemon might be old, however there really was no reason to assume he was in bad health considering that body of his. And even if there was, well I seriously doubt that they wouldn't have any alternatives for who has to lead Tootsuki. For starters it's not a kingdom, it's probably not even necessary that the director has to be someone from the Nakiri family. Normally it should be easy to let another person hold the position of director while the school remains property of the family. Senzaemon also has at least two confirmed children: a daughter that married Azami and a son that married Alice's mother. So you really can disregard the possibility that they were afraid of a power vacuum.
The E10 already had extensive powers. I mean they were even able to remove the director from his position, so in that regard their power might even have been lessened as I can hardly imagine Azami wouldn't try to make it impossible to remove him from his position the same way as they removed Senzaemon. They also already had quite a bit of fame and they're still young. Being a member of the E10 is an almost guarantee of becoming famous after graduation and earn a lot of money. Not billionaires of course, but that's really not the case right now either. In the end they are only 3 years on Tootsuki, so don't tell me they sacked Senzaemon to become famous just a bit sooner. This possibility can be largely discarded likewise.
You don't really believe yourself that Azami has gotten so far by mere blackmail, do you? Like it or not Azami has obtained a good reputation when it comes to education and results, which you really wouldn't get if the only thing he did was blackmail.
As I said most likely each person who voted for him has his or her own reasons. Tsukasa clearly believes that following Azami would be much more beneficial for the further development of his cooking abilities. Eizan then looks at things from a business perspective and thinks it's financially more lucrative to support Azami, though he already has enough money. At this point he's more of a guy who does it for the act of gaining money rather than the money itself. Rindou on the other hand seems to be doing it just because it's fun.
Well as it stands now we really don't know that much about each individual E10 member, but I think there's little doubt they all followed Azami of their own free will.
