We already know that genes regulates speech - for example, FOXP2 [0] - and have successfully sequenced the human genome, and have started similar initiatives on other archaic human and primate species.
Phylogenetic Analysis has been fairly successful already in analyzing our genetic history, so I'm not sure why you'd think it's impossible.
>As a quick point and this is quite a late response, having the same FOXP2 gene may not be enough. New evidence suggests
>Using statistical software that evaluates gene expression based on the type of gene, Vanderbilt graduate student Laura Colbran found that Neandertal versions of the gene would have pumped out much less FOXP2 protein than expressed in modern brains. In living people, a rare mutation that causes members of a family to produce half the usual amount of FOXP2 protein also triggers severe speech defects, notes Simon Fisher, director of the Max Planck Institute for Psycholinguistics in Nijmegen, the Netherlands, who discovered the gene.
Phylogenetic Analysis has been fairly successful already in analyzing our genetic history, so I'm not sure why you'd think it's impossible.
[0] - https://en.m.wikipedia.org/wiki/FOXP2