you are on the right track - I'd only add that i would try to opt for a software solution developed specifically for genetic data.

your response implies a good deal of understanding on this subject, but for completeness, the theoretical underpinnings for this were laid out mostly between 2008 - 2015, with many pubs showing increased statistical power by retaining those samples but including the genotype matrix as input into a GLMM, which allows for genomic control of relatedness by leveraging shared loci other than the disease associated one(s).

I think you will find this manuscript helpful: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3989144/; it discusses several packages that have been used to do this (EMMAX, FaST-LMM, GEMMA, GRAMMAR-Gamm, GCTA). it also provides accessible explanations as to why this works alongside a rigorous presentation of the underlying mathematical structures that power such GLMMs.

if this does not help, please reply and ill take another look.

VAL