I'm referring to the system of Old Hollywood whereby major stars would be associated with/contracted to major studios and do pictures exclusively, or pretty much exclusively for them. This system seemed to stop functioning and nearly killed the studios by the late 1960's and early 1970's. Why?
I think you have it somewhat reversed. It wasn't so much that the star system killed the studios, but more that the death of the studios' power (because of bloated productions, competition from TV and their lack of understanding of the changing tastes of viewers) which helped kill the star system.
Of course, by the 1960s, the star system was pretty much dead in the water anyway as stars had already taken advantage of their increasing power to negotiate their way out of contracts in the 1950s, weakening studio control. When Shirley MacLaine sued Hal Wallis over her contract in 1959, the signs were clear. That never would have happened a decade earlier.
There were other factors, of course, but the fact that studios just weren't as powerful anymore combined with the trend of stars increasingly taking to the courts over clearly usurious contracts were the main two.