The Russians were characteristically pragmatic. Why worry about building an engine capable of sustained Mach 3 when all your are trying to do is occasionally chases and shoot down an SR-71 of which there were only a handful. Just build afew hundred engines and replace them after those occasions you wear them out.
As an aside I always wondered how much of this story is true. In the lates 60s and early 70s Soviet metallurgy/forming in rocket engines was much more adept than the US — witness the oxygen rich stage combustion rocket engines that they mastered. Perhaps its not that they couldnt build a M3 turbojet as much as they didnt feel the need to — just replace them after those handful of Mach 3 engagements. Another possibility is that they used some sort of combustion augmentation in extremis. Something akin to nitrous — which was a one time “turboboost” (channeling knightrider)