You do have a point, argumentum ex silentio is a problematic argument, however I think we need to see the big picture of evidences in his time: in the time Valerianus was operating there is no trace of the old republican families - so it's make it's extremely unlikely that he was descended of such family unless we had a pretty good evidence to that.
And if there were any trace of any one of such families, how would we indentify them?
As stated here, the argument is circular; P. Licinius Valerius can't be from the original Licinia gens just because he lived too late; then, his name is no trace of the old republican families; then, such families didn't survive.
(Just for the sake of clarity, what we have here is an example of argumentum ad ignorantiam, not of argumentum ex silentio; the latter means that you can't infer you're right simply because your oppponent didn't answer).
ANYTHING is possible however with the lack of sources indicating it we need to ask is it REASONABLE that Valerianus was a descendant to republican family?
You can identify the old republican families by there names. the Roman oligarchy was a close one and thought it's wasn't official sons tend to inherit there father position (if your father was a senator it's likely that you be as well and so on) in the time that Valerianus lived we simply doesn't find members of the republican families mentioned in the sources and while we certainly don't know the name of every Roman official at the time the lack of mention in the sources did indicated that they stop to play a prominent role in politics (whatever they were extinct or simply vanished to obscurity).
Now if we will return to my original question, does it reasonable that Valerianus was a descendant to republican family? in light of the evidence about the republican families in his time and without any source which mention it I would have to say the answer is NO.
BTW argumentum ex silentio is used to indicate that something couldn't happened because the sources doesn't mention it (or in another words they are silence about it).