Obviously, it would depend on which country you’re asking.

No idea about the US, but what you’re describing has kinda been done. The PIs were hired for a set amount of time to track some politicians during the day, and were supplemented by freedom of information requests and data from public sources.

Most of the findings were what you would expect (Some parliament members barely came to the parliament, some had days with mostly political activists/lobbing/business magnate). There were a few “out there” examples, as one parliament member was doing grocery shopping etc. Thing is, this method is pretty good to figure out what politicians work for the public and who works for private interests, but it’s nearly impossible to actually uncover anything that’s even skirting on the illegal. A PI can’t wiretap or search private property.

A tangent, but In the same spirit, there’s a crowdfunded lobbying agency called Lobby 99.