but seeing that almost most ap/ad champs with mobility have more burst then immobile champ's burst
Mobility is very much like defense or range. It allows a champion to survive long enough to get their job done. Because pretty much every champion has some degree of mobility, range, or defense: you usually won't see a damage tax on champions that have mobility. You'l just get taxed in range or defense instead.
Case in point: most hyper-mobile champions are melee-range assassins that rely on glass-cannon builds to function.
Also, in terms of delivering damage: most ranged champions are not set up to win trades by face-tanking opponents they outrange.
Caitlyn will loose a face-tank fight to Tristana who will loose to Draven who will loose to Vayne who will loose to Tryndamere.
This generally sets up a lane dynamic where you can win if you're out-ranged by your opponent. You wait for a solid time to go ham and you can win. If you out-range your opponent you can poke them to death and win.
If that wasn't the case - ranged champions would be better than melee champions in pretty much every regard.
I mean, why bother giving so much burst (cofcofekkocofcof) to a champ who already has alot of mobility? Doesn't it feel, unfair...?
Objectively it's perfectly fair for an assassin to be mobile and bursty. That's what the assassin archetype is largely about.