In discussing the possibility of PCs being the Law, you mentioned GURPS Cops, but not GURPS Mysteries, which was an oversight, I think. Mysteries contains excellent material on the natures, requirements, and differences among four types of mystery story calling for different types of investigators:
- The police procedural calls for cops
- The cozy calls for eccentric geniuses
- The thriller calls for specialists
- The hard-boiled calls for private eyes
GURPS Mysteries explains why (p.11), and understanding this is very helpful to a GM in planning a mystery adventure that will suit an existing party of PCs or a party of PCs suitable for a particular sort of campaign.
The rules are guidelines of course. But if you know what they are and why they work you can then figure out how to bend them. Because I know what specifically makes cops usually unsuitable for thrillers and hard-boileds I can (and have) set up a campaign in which the adventures can be hard-boileds and thrillers in disguise as police procedurals. That in turn let me escape the problems that usually make it difficult to have the same characters figure in serial thrillers or serial hard-boileds.