Yes I’m struggling to see what relevance the putative reality of LUCA as the last universal common ancestor of bacteria and archaea, has to the GAE. Regardless of whether these two prokaryotic domains really share a single ancestral species, population, or bottleneck otherwise, and the extent to which horizontal gene transfer has implications for how “tree-like” individial gene-genealogies would have been ~three-thousand eight-hundred million years ago, it seems to me that has next to nothing to do with humans being primates, mammals, vertebrates, or even eukaryotes.
If the bacterial and archaeal domains ultimately derive from independent origins of life(and heck, even if the origination event was magic), and merely appear to share ancestry through some combination of radical levels of HGT and convergent evolution, then… what does that matter to how hairy my one hundred thousandth-great grandfather was, and if we should call him Adam?