Pilots actually do exactly what you suggest. I certainly did! Nobody wants to be a sitting duck!
However missiles are extremely high tech, maneuverable, and smart. They are programmed to intercept you, regardless of how you might maneuver. Sometimes the pilot wins; sometimes the missile wins.
There are various methods of missile guidance and control including radar beam riders, IR homers, active homing radar, etc. All are programmed to almost instantaneously counter and follow any of their target’s maneuvers, trying to evade the missile. Once the missile is locked on it is difficult to shake.
One of easiest maneuvers is against a pulse doppler radar guided missile. One only needs to turn 90 degrees to the missile/radar to disappear into the clutter of everything else with equal relative velocity to the radar/missile.
IR guided missiles are difficult to out maneuver. They will pursue you , unless you can break their lock, which can be very difficult. (Yes, flares come in very handy.)
Flying in the Vietnam War, (in addition to ECM and chaff) we used the barrel roll technique. We would maneuver our aircraft to put the missile at either our 10 or 2 o’clock position. We then waited until the ‘right moment’, to perform a “high-G barrel roll* that the Mach 3 missile could not (hopefully) follow. Since the missile had to anticipate where I would be in the very slight and fast future, changing my flight path in three dimensions rapidly and dynamically would sometimes defeat the rapidly onrushing Mach 3 missile’s capabilities. It worked for me more than once.
Here is an excellent discussion on avoiding SAM’s: