I think it’s more that those are two totally opposite extremes of “manhood.” One’s extremely macho and manly, the other’s a stereotypically gay profession for a male.
Probably that Beaver was known for getting into scrapes, messes and predicaments. Doesn’t go with the fastidiousness associated with beauty parlors of that era.
At around the three minute mark it is stated that the former professional athlete that is presenting some of the awards is now part owner of a beauty parlor.
I think that’s it. Characterizing a man as a hairdresser back then was as far as you could go in alluding to homosexuality back then.
ETA: And I see Lance Turbo watched the episode and picked up the basis for the joke. Still, in TV shows at the time, the intent was clear. A male hairdresser is a man doing woman’s work.
Thank goodness. Not understanding the jokes in Leave It To Beaver is as bad as not understanding the day’s Family Circus. It could have caused lasting damage.