No More Jockeys - CodeGolf Version
Python 3, 62 bytes

f=lambda a:0in(f([x for x in a if{r}-{*x}])for r in sum(a,()))

Try it online!
Python 2, 61 bytes

f=lambda a:0in(f(filter({y}.__sub__,a))for x in a for y in x)

Try it online!
-5 bytes thanks to Unrelated String
-6 bytes thanks to dingledooper
-1 byte if we use Python 2 thanks to dingledooper
-2 bytes thanks to xnor
Outputs True for player 0 winning and False or () for player 1 winning (() is an empty tuple in Python, which is a falsey value in the language)