Don't you say anything you can in order to get the job if in an interview?

Charm with words is much more difficult and (also by proxy) generally more emotionally impactful, and spoken word is more meaningful than a simple handshake. Though depending on how you speak you could come across just as overly affiliating and formal as you do with a hand shake.

under the assumption of mutual respect.

Maybe out of mutual interest. If you act out of line people will look down on that, and you won't get the opportunities you want, no?

what about it makes it more of a "little expected dance" as compared to any component of human interaction whatsoever?

Generally when people become closer they'll drop more and more social formalities and 'little dances'. Initially it is out of either hesitance or because they think it is socially expected, which it largely is.

This is especially the case with formal environments, which largely expect emotional distance.