> Why anymore so than if YOU go to a strip club?
Two reasons:
1) People in the spotlight (like politicians) are expected to behave as role models. Especially where moral values are concerned, one can reasonably demand that politicians live according to the same standard as they demand from the citizens. Otherwise, it rapidly devolves into "rules for thee but not for me", destroying trust in the fairness of democratic institutions.
2) In close conjunction with #1, behavior like cheating on one's wife or other (mostly sexual) deviance is either considered socially inacceptable or at least to be damaging to relationships. This creates leverage for bad actors, known as "Sexpionage" - the "kings" here of course are the Russian KGB/FSB who are at least rumored to have actual material of the current US president.
Basically, the public should demand transparency and decent behavior from politicians and other state officials to make sure they are not subject to extortion attempts no matter by foreign states, criminal organizations or by legitimate mega-corporations.
> Because one SUPPORTS a specific policy, does not mean one has to LIVE their life by that policy.
The more radical you appear, the more scrutinized your behavior will be. No one's gonna bat an eye about the contents of your trashcan if you're just supporting a law mandating recycling... but if you are the major initiator of a law or lead public campaigns for recycling, prepare for being in the spotlight.