"It isn't practical to do" and "society at large didn't go this direction" are very different statements.
Hyphenation is two names in a trench coat. Maintaining two names indefinitely works just fine as long as you discard rather than endlessly compound. Presumably the only requirement is that it be straightforward to trace any given lineage.
The traditional approach is for women to keep their maternal name and discard their paternal name on marriage while men do the opposite. But of course any scheme could work, up to and including each person arbitrarily choosing which name to discard (not sure how they decide on ordering in that case).
Another historical approach is the Foo Barson, Baz Fooson (Barson) approach. That scheme treats the male and female lines as being entirely separate so it doesn't quite match what you're after but it was quite practical.