Genes do explain much of the variation between people for many characteristics. You can think of them as initial character stats that are then modified by environmental factors and random noise.
It's important to note that heritability is not some kind of magic bullet that explains everything with "genes". If you want to convince a biologist that "genes" are the cause for a given phenotype, you've got to show an actual genetic causal mechanism or you've got nothing. This is, again, due to the myriad of interfering mechanisms that make "genes" very, very removed from the actual phenotype. And again, I'm not talking about the environment.
One of which is roughly DNA that codes for a protein, or controls such, and indeed the causal mechanisms from there to phenotype can be extremely complicated and hard to figure out.
In the context of "variation between people" there's another meaning, of just whatever it is that causes inheritance. We could say many things about this (e.g. from twin studies) even before we'd discovered DNA, and non-DNA routes (like if most babies get some microbes from their mothers) would still count.
Your second point is not true for complex polygenic traits like height. Height is strongly influenced by genetics but we will likely never understand the exact mechanisms that do this.