It is my understanding that these aircraft rarely fly at supersonic speeds anyway—the ordnance they carry, the increased fuel burn (even with supercruise), and the increased chance of detection make supersonic flight either hard to maintain or seem unnecessary. In addition, it would seem more beneficial to perform subsonic evasive maneuvers against an incoming missile than to try to outrun it (many air-to-air missiles fly at Mach 3 or higher). Finally, the subsonic Sea Harrier even downed a supersonic-capable Mirage III and many supersonic-capable IAI Daggers to no losses during the Falklands War. Though this can be attributed to the superior pilot training and offensive technology of the Harriers, how come many other aircraft were/are produced with a supersonic requirement?
One reason is interception (with interceptor, not missiles).
Defence is probably more important. Defence with fighters is still important for three reasons:
- There are relatively effective counter-measures against surface-to-air missiles. A fighter pilot can adjust the strategy to the counter-measures encountered, missile can't.
- In peace or limited conflict situations it is often not known whether a suspicious target is hostile at all and the only way to identify it is to have a fighter fly to it and have the pilot identify it with their eyes.
- Fighter can also do other things than use lethal force. In peace time you generally don't want to shot down an intruding military aircraft of another country straight away. Fighter can chase it away from your territory or can try to arrest it (force it to land on your own airbase). And fighters are also used to identify and possibly navigate to safety civil aircraft that lost radio contact.
So when a potentially hostile aircraft is detected entering your territory, fighters are sent to intercept it and help it, arrest it or shot it down depending on what it turns out to be.
If the fighters are slower than the intruder, they may not be able to intercept it at all (they don't usually know where it is flying so they can't wait for it). The faster they are, the larger area they can defend. In this case the interceptors take off without external fuel tanks (as they won't be flying very far) and with only relatively small air-to-air missiles, so they can fly fast.
Conversely if you try to penetrate the enemy territory, flying faster limits the number of fighters that can intercept you. However in this case it is more useful to rely on stealth or flying low to evade detection, so attack aircraft often are either not supersonic or have lower maximum speed.
The answer, speed is life. While that may seem abbreviated, superior speed gives fighter aircraft far more options to attrite the enemy or to survive another day.
The following is a limited subset of general reasons why having superior speed would be an advantage, but unfortunately, specifics are beyond the scope of this discussion.
Offensively, speed is necessary to consummate an intercept, although, it often does not require superior speed, as geometry is a far more useful intercept tool. However, offensively, speed becomes more useful when chasing down a fleeing aircraft, or when flowing to multiple targets.
Defensively, speed can help targeted aircraft bug out, as well as simply increase time to kill so that other friendlies can swoop in and save the day. You burn far more gas exploding in a fireball than you do dumping fuel into the afterburners!
However, the power to go fast is often just as important, if not more important that the ability to go fast. Those afterburning, fuel guzzling, engines allow fighters to have enormous rates of climbs, and to sustain airspeed through much more dynamic maneuvers than slower, less powerful, aircraft. Thrust to weight ratio is paramount in a turn rate fight, and, in BFM, the ability to suddenly add a fist full of knots may be the deciding factor between missile giver and missile taker.
To add to other answers, aircraft speed and altitude greatly contributes to missile kinetic performance. The range difference between a missile fired at mach 1.6 at 35k feet versus the same missile fired at mach 0.5 at 10k feet is (tactically) massive. Also faster fighters will generally climb faster while covering greater distance which is important for interception from ground scramble.
Another Addition to the previous answers would be: Most maneuvrability/performance theories are based on the Energy of the aircraft. So the higher and faster you fly, the more Energy you have for combat. You can convert this speed in altitude for example. This was especially true for military aircraft in WW2. For example: https://en.wikipedia.org/wiki/Energy%E2%80%93maneuverability_theory