Worth observing here that military aircraft having engines with cooling problems and consequent engine life issues is nothing new.
The B-29 that dropped the atom bombs in Japan had engines that were apparently so prone to overheating that rather than prioritising gaining altitude after takeoff B-29 pilots had to prioritise gaining airspeed lest the engines catch on fire. The engines also had an extremely short service life, with engines replaced after as little as 75 hours of service [1]
Similarly, the Soviet MiG-25 was theoretically capable of well over Mach 3, but was limited to Mach 2.8 to prevent the engines overheating [2]