I'll say the same thing as you: context matters. Someone trying to say that smarter parents lead to a smarter student body doesn't need to model any genes and they don't need to care about the difference between things that are transferred genetically and things that are transferred socially.
> because of genetics as the main determining factor?
Does that matter? While the word "heritability" was used, and that term "very much has to do with genetics" as you say, that person didn't directly mention genes and didn't attribute any particular percent to genes. The original argument is the same whether genes are 80% or 20%.