No. ANW has in their NDA that it will not allow current ANW competitors on Beastmaster. They are competing shows. Beastmaster is its own show and doesn't belong in ANW. ANW belongs to NBC and Beastmaster to Netflix. So far the only ninjas there are not current or not from American Ninja Warrior (but their spin offs which do not have the clause in their NDA). Others have tested it as part of the ATS crew. However, you will not see them competing on the show unless they have no desire to return to ANW. And finally, the producer Dave Broome is trying to make sure both shows do not overlap. Beastmaster is unique and there is no point in muddling the waters by adding ANW competitors. This is not a spin off.
In terms of the "family tree" of the actual show's history, it is completely unrelated to ANW.
The closest thing it has to a relation is actually in the foreign competitors. If I recall correctly there have been a few people from Europe / Australian Ninja spinoffs that appeared, and in S1 Team Japan had 6 people who have appeared in Sasuke or KuroOvi.