Well, Rogers was actually a bit of a Renaissance Man: university-trained composer, ordained minister with a specialty in counseling, and executive in his family's oil-tool company. Plus, of course, he was a chlidrens' entertainer who had interned in the big leagues at NBC in his younger days.
So, it's not entirely ASB to imagine that he'd take an active interest in politics. And in fact he did have some experience with
political activism.
But the presidency? I dunno. First, you'd probably need to get him into Congress, likely the House. And, while others can correct me on this, I'm not sure if Pittsburgh would be the safest place for him to run. More like some diehard liberal enclave somewhere, populated by Great Society bureaucrats with maybe a smattering of bohemian youth and college kids. Of course, he runs as a left-leaning, pro-spending liberal, albeit one insulated from the usual caricatures by his straightlaced Middle American image.
I'm not sure how you get him from Congress to a place on a presidential ticket. Maybe a year when Democrats figure they're fated to lose, so they might as well shore up their liberal base and toss in some star power at the same time, and ask Rogers to join the ticket as veep. But the GOP unexpectedly screws up their campaign, the Dems score an upset, and the POTUS dies some time during his tenure. Rogers assumes the presidency, and "Can you say deficit spending, boys and girls?"(*) becomes a staple of right-wing humour.
(*) In fact, I don't think Rogers ever used that "Can you say...?" line on his show, but it's sorta become his version of "Elementary, my dear Watson".