For the Viennese classicists the only real rivals were Italian opera composers, especially Rossini. Otherwise, they were very much the highest standard in music of the time. But Schoenberg and his students faced considerable competition from other aesthetic and technical approaches. They were not the only game in town. The most important other schools of thought were the modern French composers such as Ravel, Debussy and Messaien, the Germans who were still writing tonal music such as Richard Strauss, the Hungarians such as Bela Bartok and a few isolated figures like Jean Sibelius, the great Finnish symphonist. But the biggest rival was the Russian Igor Stravinsky who had achieved great fame and popularity while still retaining his modernist 'street cred'.
There are a few strange things about that piece that I will take up in my post on Schoenberg...