D&D without a fighter

Imagine a D&D without a fighter class? Imagine it without all the problems it brings! No more having to balance a game around a class that shouldn't exist. No longer an exponential vs linear progression issue. Gone solved, done for. The fighter is an extinct species.

You know when this hit me? As I was thinking about dungeon clearing techniques I realized units like SWAT, Rangers, Delta and SEALs are not fighters, they're thieves, magic user thieves for that matter (or thief magic user, whichever way you want it). The modern day "fighter" can jump off an airplane, swim underwater, climb up and down a rope, listen through walls, speak in sign language to fellow thieves err team members, break through doors, pick locks, blast doors, sneak and attack, sneak and back-stab, move silently, see in the dark, attack from afar, throw grenades, etc. More so one in four or one in eight are cleric thieves. They can do all that cool stuff and heal too. What can the fighter in D&D do? Swing his sword? Woooaaahh wow!!!

Why not empower all classes to have fighting abilities to a higher or lesser degree and be done with the fighter as a class? Fighter is now a skill and not a class. Something you can add to other classes to a greater or lesser extent.