San Francisco Ballet School is not a classical training academy: like most North American elite and company-affiliated schools, the training is an eclectic mix designed to feed dancers into North American companies, which, with a few exceptions, perform a wide range of rep, full-lengths plus short neoclassical ballets. Gloria Govrin was associate director of the school for nearly a decade, and she was a Balanchine dancer. Kirov Academy and Goh Ballet are classical academies in the US. I'm not sure where Houston is now, and Harid might also have more of a classical emphasis. There are smaller schools where classical technique is emphasized or even taught exclusively, but even these students mostly end up being "finished" at a company-affiliated school if they don't get hired in Europe and gain experience there.
Pennsylvania Ballet's only Asian-born dancer trained at SAB. At PNB Kaori Nakamura, who retired last year, was trained in Japan, but chose SAB from her Prix de Lausanne prize. Sokvannara Sar's only ballet training was at SAB after being given a crash course in ballet. There have been other Asian-born dancers at PNB -- Le Yin went through the state-sponsored training in China and Batkhurel Bold was trained in Perm -- but the majority of dancers of Asian descent in the company are US-born and -trained.
Most North American companies are just as focused on neoclassical as classical ballet, and often dance classical ballet in neoclassical style. It hasn't stopped some from hiring Asian-born and -trained dancers into their more eclectic companies: Royal Ballet has five out of over 100 dancers, San Francisco Ballet has eight out of 76 dancers, National Ballet of Canada has ten of 75 dancers, and Ballet Arizona has three out of 31 dancers. In the North American companies, I don't see the dancers being pigeonholed into the classics and not cast in neoclassical works.
Companies tend to reflect their audiences; the very smart ones also reflect their community.