The OP's characterization of it is wrong--three non-violent felonies couldn't trigger three-strikes' even before California's recent reform (you had two have two violent felonies first).
Moreover, for-profit prisons barely existed in the early 1990's when states like California and Washington started passing three strikes laws. Law enforcement, prison unions, private prisons, etc, have definitely benefited from these things, but the actual policies arose from public outrage in the 1980's and 1990's about the government being "soft on crime." For example, either California's law was passed on the back of public outrage at the murder of Polly Klass.
One bit of context that's lost in current discussions is how much consternation existed in the 1980's and 1990's about crime. If you look at the media from the time, there is a ton of hand-wringing about criminals being let out early on parole or clearly guilty people getting acquitted "on a technicality" (usually a 4th amendment violation).
A movie that captures the zeitgeist of the time is the 1983 movie Star Chamber: https://en.wikipedia.org/wiki/The_Star_Chamber: "Judge Steven Hardin (Michael Douglas) is an idealistic Los Angeles jurist who gets frustrated when the technicalities of the law prevent the prosecution of two men who are accused of raping and killing a 10-year-old boy. They were driving slowly late at night and attracted the suspicion of two police officers, who wondered if the van's occupants might be burglars. After checking the license plate for violations, the policemen pulled them over for expired paperwork, claimed to have smelled marijuana, then saw a bloody shoe inside the van. However, the paperwork was actually submitted on time (it was merely processed late), meaning the police had no reason to pull over the van and Hardin has no choice (see fruit of the poisonous tree) but to throw out any subsequently discovered evidence, i.e. the bloody shoe. Hardin is even more distraught when the father of the boy attempts to shoot the criminals in court but misses and shoots a cop instead. Subsequently, the father commits suicide while in jail only after he informs Hardin that another boy has been discovered raped and murdered and tells him "This one is on you, your Honor. That boy would be alive if you hadn't let those men go." After hearing all this, Judge Hardin approaches his friend, Judge Caulfield (Hal Holbrook), who tells him of a modern-day Star Chamber: a group of judges who identify criminals who fell through the judicial system's cracks and then take actions against them outside the legal structure."