With more DNA data, we wouldn't necessarily need linguistic characteristics to chart language ancestory, we could also look at the DNA evidence.
> one reads Homer, the Rig Veda, and the Twelve Tables side by side, one gets the distinct hard-to-articulate impression that these texts were produced by closely related cultures.
We know that the romans borrowed heavily from the greeks. That rome and greeks were closely related is well known. Everything from law to literature to religion. Not sure about the Rig Veda.
> I could point to discrete things like how patriarchal they were (even by ancient standards), the importance attached to herding and poetry, etc.
That describes a lot of cultures.