The last major attempts for a "one size fits all" muti-role fighter was the f4 which resulted in the services abandoning the approach in favor of the F18, F-15, and A-10. Like a bad penny the multi-role fighter concept just keeps coming back. We are ending up with a plane that does everything and will not be able to do any of it particularly well.
I see you've conveniently forgotten the F-16, which is the plane the F-35 is succeeding. The F-16 has been a resounding success. Whether that translates to the F-35 remains to be seen, but the precedent is there.
The F-15 and F-18 are also highly successful examples of multirole aircraft, FYI.
I do wish the F-22 production line hadn't been shutdown, it could have been a very successful export to Japan and Australia...plus the US could have bought a few hundred more.