I don't believe that using the nominal modeling type would cause a singularity in the CMH tests, especially since you have only two levels for X and for Y.
Yes, the null hypothesis is no association between X and Y.
Yes, the unstratified contingency analysis is highly significant (LRT chi square > 65, p-value < 0.0001). (Note that chi square cannot be used to assess the strength of the association, only its significance.)
Yes, the relevant CMH test (General Association of Categories) is highly significant (chi square > 40, p-value < 0.0001). The stratification helps assure that the association is consistently present across the strata, if that question must be answered. The CMH test also somewhat relaxes the sample size requirement for the chi square statistic, which is not an issue in this case. (No more than 20% of cells with expected count less than 5.)