doesn't the lottery of meiosis randomly give you genes from each pair of grandparents, thus ending up with one random maternal grandparent choice and one random paternal grandparent choice at each position pair (maternal contribution and paternal contribution) in each cell (of course recursively happening during creation of haploid germ cells within each person) ?

Yes. But if you're suggesting that you could treat differences with cousins as random, the way we can treat differences with siblings, then no, because of assortative mating; e.g. if my cousin's "good genes" came from my uncle, then maybe he married my very rich aunt who left my cousin a large inheritance.

Due to chromosome crossover, you do not inherit genes individually from ancestors, but instead you inherit segments of chromosomes containing many genes.

However, the randomness is great enough that first cousins do share approximately 1/8 of their genes, second cousins 1/32, third cousins 1/128 and so on. The rapid decrease of shared genes among relatives in successive generations also means that if mating in a population is really random, the population rapidly becomes homogeneous.

It is only assortative mating which generates and maintains distinctive subpopulations.