If we look at some other physical activities we can see variations that qualify as cross training that appear to be pretty successful.

Gymnasts compete in multiple events demonstrating different skills. Dancers usually train in more than one form of dance. Decathletes and pentathletes train in different events utilizing different skills.

Warriors in Japan and and Europe trained in different weapons, not just one.

If a person trains kicks, strikes, throws, falls and few weapons, even within a defined Art, THAT is cross-training!