The wikipedia page on gens notes that
Although both the concept of the gens and of the patriciate survived well into imperial times, both gradually lost most of their significance. In the final centuries of the Western Empire, patricius was used primarily as an individual title, rather than a class to which an entire family belonged.
The gens originally held a governance function, then that governance function was absorbed into Roman governance. After the Imperial period, the functions were no longer relevant. There was no reason to maintain the lineage, and the "notable families" bred back into the general populace.
With respect to Latin surnames, I don't have any data on the distribution of surnames, so I don't know if Latin surnames are under-represented. If I were to hazard a guess, I suspect that after the fall of Rome, the prestige of a Latin surname vanished, and I would expect that today Latin surnames are probably similar to other ethnic surnames in distribution - but that is merely a hypothesis and I don't have data to test it.
Update: Even when Rome controlled Europe, only a minority of the population would have had
a nomen. I cannot cite a source, but I believe that the majority of nomen were from the Urban Tribes, and therefore would not have been "notable". The governor of a province and his staff probably had patrician nomen; the inhabitants of the province probably did not. Again, a hypothesis, but I would be surprised to find surnames derived from nomen anywhere other than Italy, Byzantium and possibly Spain/Portugal.