Anyone could have, by the time Nixon went to China, China had only distant Albania as a reliable ally. As for the Sino-Soviet not being public, it didn't matter, even when the U.S knew of the split they considered China a bigger threat since the Cuban Missile Crisis in the early 60's, because the Maoists actively supported conflict, unlike Khrushchev for all of his bluster actually supported peace coexistence. Hell by the Vietnam War the U.S knew of the split but believed that if they left it would validate Mao and cause the Soviets support more conflict to save face. As for the Split not being public, you had lots of print literature by many people who studied Communism that speculated the hell out of the Sino-Soviet split.
Only when Le Duan despite being backed by Mao made more overtures to Brezhnev for aid, who wasn't as much of pain in the ass to work with unlike Mao, did Mao consider working with the U.S. Unless you can get a non-Maoist Communist that's not aggressive, then of course anyone can go to China as it were, so Nixon wasn't that special in my opinion in that case.