Dunno about you, but the Czar always liked Leave It To Beaver, and frankly thinks there are two types of people in the world: those that make fun of it, and those who actually watched an episode or two.
Beaver portrayed a show that a lot of people thought was unrealistic; however, the show was quite plausible. Kids would get into trouble through their own sense of adventure or overconfidence, recognize their mistake, and try to cover it up. Parents would eventually piece together what happened and attempt to deal with it in a rational way. Not all of us grew up that way (the Czar sure did not), but many of us did. And those who did not could look to the show and figure that out.
Plus, what many people conveniently forget, the show was freaking funny. The kids twisted logic, the exasperated parents just trying to get to the truth, the nutball personalities, and the clever wordplay resulted in a show that did not resort to Very Special Episodes, weekly celebrity walk-ons, cross-marketing tie-in appearances, Cousin Olivers, or spin-offs with the wacky neighbors.
Ms. Billingsley as the confident, capable mom who could handle a crisis as well as pull off a great double-take, was absolutely a major part of the shows success; her character was not a one-dimensional stock 1950s mom, but represented what a lot of moms were: competent and cool, smart and quick with a good line, bright, and able to fix stuff. It is hard to imagine that didnt originate with Ms. Billingsley herself.