What exactly is not naturally sefardi about anything associated with shas? Sefardim don’t seem to look at it that way, as almost all of them support it. As far as i know, the only prominent sefardi rov who didn’t support shas is rav yaakov hillel.
They build yeshivos which teach sefardi mesorah; i have no idea what you can possibly mean by saying that they’re not sefardi.
Maybe because you think that “natural sefardi” equals an almost illiterate farmer who kisses the hand of the mekubal? Pity if you think that’s all a sefardi person is capable of.