If you assume a fixed probability distribution over possible $u_y$ that both players know when coordinating, then they can set up the rules they choose to make sure that they probably win. The extra random information is only useful because of the implicit “for all $u_y$”. If some malicious person had overheard their strategy, and was allowed to choose $u_y$, but didn’t have access to the random number source, then the random numbers are useful.
If you assume a fixed probability distribution over possible $u_y$ that both players know when coordinating, then they can set up the rules they choose to make sure that they probably win. The extra random information is only useful because of the implicit “for all $u_y$”. If some malicious person had overheard their strategy, and was allowed to choose $u_y$, but didn’t have access to the random number source, then the random numbers are useful.