In addition to the economic and practical limitations noted (how many murder cases are there in which there is a willing client who would even be able to, much less willing to, pay what it would cost for a detective to duplicate/go beyond the police routine?), there are a couple of other problems.
First, as noted somewhere above, “solving” the crime implies getting a confession or conviction. How’s a private citizen going to do this? In the Nero Wolfe stories, Wolfe typically solves the crime through a series of mental conjectures, then summons all the suspects (who are always, conveniently, acquaintances of the deceased, rather than, say, itinerant carjackers or street people) to his house, confronts them with the evidence and (often using a psychological ploy based on a conjecture that he believes to be true, but can’t yet prove) goads the guilty party into confessing or otherwise revealing himself.
Well, in real life the guilty party has a lawyer who is not going to allow him to attend such a gathering, and guilty parties often maintain their denials in the face of overwhelming hard evidence against them, let alone a chain of logical but unsubstantiated guesswork presented by some P.I.
Wolfe also had the tacit if fractious cooperation of the police (they would grudgingly show up at his climactic denouements, albeit disclaiming official sponsorship). In real life, at least today, I’d think cops would be reluctant even to accept any evidence generated by a freelancing P.I., let alone place themselves in a position where they could be viewed as endorsing his acts. At least in the books, P.I.s are often effective because they are, um, willing to be more creative and less hidebound than the cops, which translates into not following all the silly procedural rules the cops do, if skipping this procedure can help them get to the truth. In the post-Warren-Court era, a competently-advised police force would need to be vary wary about accepting, let alone relying on or endorsing, evidence generated by a P.I. using some unknown magical methods – at best, it may be thrown out at trial as “fruit of the poison tree” if the P.I. obtained it by, e.g., housebreaking or other chicanery. At worst, accepting such evidence could be seen as sponsoring the P.I. or using him as a state agent in his “acting under color of law” such that any improper or abusive behavior of the P.I. could be imputed to the P.D. and form the basis for the suspect to assert tort liabilty against the police.