The USAF operated (has been operating? heh heh) MiG aggressors for decades.
Google "Constant Peg" -- just one declassified of many still-classified programs doing this. Have Drill, Have Donut, Have Idea, etc, and many before that, too.
The idea with the Flankers is that a private contractor can do it cheaper and better, and not have to deal with any of the hush-hush sourcing issues with the aircraft. One of the reasons for the secrecy (there are many, of course) is that the countries who supply the iron to the US don't want other countries to know about it. With deals like these Flankers purchased on the open market, there are no issues with this. There are also no issues with other countries knowing exactly what type/variant/equipment/weapons are operated by the company. There is a big cost savings by dropping big portions of the immense security blanket that goes over black programs.
When the Moldovan Fulcrums were purchased by the US in the 90s, Chrysler Aerospace and at least one other major company (Raytheon, maybe? My memory fails me) were trying to start up a "commercial aggressor" operation with them, but never actually came to fruition.
