I was literally just listening to the comedian Steve Harvey comment on how "barbershops are the poor man's country club," ... and this is precisely the theme of Leacock's (more ancient) piece as well. According to Leacock, men go to the barbers to be schooled on the information of the day, especially about athletics.
"Men Who Have Shaved Me" in Literary Lapses is comedic in tone and about contemporary education (street smarts?) in theme. First, Leacock talks about how barbers (these "men who have shaved" him) are experts in athletics. Basically, the customer is to be, basically, hog-tied ...
... and then given such necessary information on the athletic events of the moment as will carry them through the business hours of the day without open disgrace.
Leacock has no problem admitting that men everywhere wait long hours to submit to this very treatment. The information is not only "important" but "critical" to one's social standing. You can hear the humor and satire here. The irony, says Leacock, that the barber doesn't really want his customer to talk much. Why? Because the barber would prefer to speak only with experts: other barbers. Further, Leacock jokes about the way barbers are experts in "Physiology" and "Chemistry" and "Practical Anatomy" in humorous and barber-related ways. Another irony is that the more the customer is manhandled, the more the barber enjoys his job, ... while imparting said information.
In conclusion, we should say that Leacock would disagree with Harvey that barbershops are just for the "poor man." In fact, Leacock would say that the barbershop is for ANY man who wishes to be well informed.