nobody said bruce lee invented cross training. the samurai were cross trained, technically. These days, when people mention cross training, it's the thought of training striking and grappling. In tma, you don't really see that due to the lack of grappling in cma in general (I'm not talking about chin na and shuai chiao, but ground grappling). you saw people training clf and lama pai, longfist and taiji, etc. multiple standup styles, which yes, is cross training. But, as stated that's not the general line thought of today when cross training is mentioned. boxing and wrestling is thought of as cross training. mantis and judo is thought of as cross training. karate and wing chun is not.
that said, CMA made it taboo on their own, I'm guessing. ever since the advent of the ufc, I've heard several tma say "we don't need to learn grappling," followed by some reason or excuse, ranging from they have grappling in their system to they won't be taken down.
i'm nobody...i'm nobody. i'm a tramp, a bum, a hobo... a boxcar and a jug of wine... but i'm a straight razor if you get to close to me.
-Charles Manson
I will punch, kick, choke, throw or joint manipulate any nationality equally without predjudice.
- Shonie Carter