I hate the "it's old, ergo it's bad" line of arguments. His hygiene argument, which seems to be quite secondary, is also bad. Being in the same room as someone who sneezes will also get you his germs.
The solution is not to end the medieval, heathen practise of interacting with other people in real life, but to stay at home when you're sick - parallel to keeping your hands clean, doubly so if you expect to be shaking hands.
I think he makes quite clear that the main argument is that it is not hygienic. In fact his argument that it is an old practice plays into to his hygiene argument. I think what he is saying is that the practice was developed before the germ theory of disease; before people truly knew how disease was spread. With our "updated" knowledge, we should have an "updated" system of greeting. So, though I would usually agree with your point about hating "it's old, ergo it's bad" types of arguments, I don't really think it applies here.
I completely agree with you. Lets think of a few things which are also medieval or even older! Well there is farming, should probably stop that, oh and preserving food, should probably stop that too.
What the author seems to have completely missed is the social bonding and the shear volume of information one can share in that second or two of holding hands.
The solution is not to end the medieval, heathen practise of interacting with other people in real life, but to stay at home when you're sick - parallel to keeping your hands clean, doubly so if you expect to be shaking hands.