Do either of these probiotic supplements have antiviral activity and how important a contribution might that be to their efficacy?

These do not have antiviral activity per se, but the creation of a healthier oral cavity through diminishing the harboring of other microbial species has some prophylactic advantages. The main way in which viral intruders can be kept at bay is the mineralized barrier which will, in fact, reduce the ability of viruses to infect the tissues. By reducing bacterial invasion of the gums, for example, there will be less damage that opens up the tissues to further invasion by viruses. Similarly, the advantages of promoting more shallow pockets that provide a safe harbor for microbes to fester and invade the gums will also have a prophylactic benefit with respect to viral invasion, to some extent at least. So this complex picture is one of having a number of factors working in your favor and that is the reason for using such supplements to begin with. Nothing is perfect when it comes to nature, but some things are more robust and effective than others in promoting health and well-being.