Does that mean the police are allowed to invade my privacy like that if they're just "checking for potential illegal activity"?

No, just that you cannot use a statute like the peeping tom one against Law Enforcement in the same way you'd use it against an average joe.

But also, yes, the police can do basically anything and receive "Qualified Immunity" as actors of the state. Under the current interpretation of qualified immunity unless the police know what they are specifically doing isn't allowed then they have immunity, and the court system gets SUPER specific with the facts to the point that pretty much any action gets immunity. The state itself might be in trouble but that's a whole other mess of immunity.

I'm not a lawyer though and I'm almost certainly messing up some of the nuance here.

Qualified immunity is a creature of federal Constitutional law; it exists to immunize law enforcement from civil suits alleging that they violated someone's Constitutional rights. The doctrine does not immunize officers against criminal charges, particularly crimes against the laws of the States.

"Privacy" isn't (yet) a Federal civil or constitutional right so I'm not sure that qualified immunity would apply. The intent qualifier in the Peeping Tom law is probably sufficient since the intent is not to violate someone's privacy but to follow up on some sort of reasonable suspicion of criminal activity.

Also if people can see through your window doing ordinary (non-traditionally-"private") activities, there's the "plain view" doctrine -- although that's related to admission of evidence and not a prima facie criminal law violation. Nevertheless a Court is probably going to use similar reasoning when divining intent.

Thanks for clarifying! My knowledge of the law is second-hand. Comments like this help a lot!