Yes there is a genetic components to LADA/ T1 . Having a genetic susceptibility does not mean you will have T1, in your case it did, but generally it just increases your risk. Several genes have been identified as influencing a gene response, they are the HLA DQ8 and DQ2 and HLA-DR3/4/5 alleles. Also having a first degree relative with Type 1 further increases risk , father, mother or sibling. Having another autoimmune disorder increases risk. All these factors increase risk, but not inevitability.
For instance, having a genetic propensity only, the risk factor is low, but having a first degree relative with T1 increases your risk of developing T1 by 20x . Having another autoimmune disorder adds further to the risk.
In point, I have all those risk factors, the genetics, the first degree relative and an autoimmune disorder. I am also producing low levels of one of the autoantibodies associated with T1, again that’s another risk. I have not , at this time, developed T1, I do have a life time risk of developing it, but it is not an inevitability.
Edit - I’ll dig out some info for you on risk