What will be an appropriate mathematical distribution for SNP data?
Simulating genotypes with realistic correlation structures is indeed not so simple, and there's quite a few papers dedicated entirely to that (e.g. https://bmcgenet.biomedcentral.com/articles/10.1186/s12863-015-0173-4). Also, DEPICT (https://data.broadinstitute.org/mpg/depict/index.html) comes with a number of simulated GWASs to generate the nulls, so that's a simple way to obtain some ready-made fake data.

On the other hand, indepedent SNPs should behave as draws from $Binomial(2, MAF)$. Note that the standard QC procedure of filtering out genotypes that deviate from Hardy-Weinberg equilibrium is just a goodness-of-fit test against $Binomial(2, \hat{MAF})$. If you employ this filter, some LD-pruning, and still get different results, consider posting more details about the discrepancies you see - that would be indeed unexpected.