As a player, it strikes me as intensely unfair to AP mages to have these other "mage" characters who can then just punch you in the face should their abilities fail. (Or, just as egregious, heal off damage after a skirmish via Lifesteal while most other mages just have to suck it up or recall.) And as someone who enjoys game theory, it seems mechanically inconsistent - breaking what feels like one of the core rules of the game - for no apparent reason.
The game-design reason?
Range.
Barely any AD casters actually have much range, the vast majority of them are fighter and assassins with a reach of 125.
Short-ranged mages like Ryze still reach 700 units with the bulk of their kit, and mages are also home to the longest ranged, strongest poke champions in the game (
).
With a significant lack of range, melee champions just need something in their kit to offset that disadvantage.
With original fighter/assassin champions like Tryndamere and Yi, Riot tried giving them an absurd level of AA DPS, but that came with massive design issues in that the damage potential of ranged champions is so high as-is going overboard seems excessive, and placing that Phys damage all in auto-attacks just doesn't serve the game well. Auto-attacking really hard is brutal in a 1v1 duel, where such sustained DPS is easily negated to 0 in the midst of a team-fight, where burst would better serve the assassin/fighter-assassin/fighter's intended combat movements.
Hence - AD scaling on abilities.
This allows more overall damage than ranged classes due to how armor and MR items are set up (as armor items hit physical damage and auto-attacks, and magic resistance items tend to hit magic damage and abilities). AD casters innatly bypass 1 of the 2 qualities of most defensive items, giving them effectively more damage to match the significant drawbacks of being melee, but without that damage going crazy overboard. Their QWER bases are often the same as mages and marksman, they're just slightly tougher to counter-build.
Also - setting up their damage in more cooldown-heavy places over sustained DPS allows them to have a more even performance between duels and team-fights, maintaining the QWER burst necessary to function while wadding through a barrage of AoE slows, while still having bursts of power you can try playing around in a 1v1.
As for the life-steal bit: a wide number of AP scaling champions can lob things from such a distance they have little reason to need sustain, as they can pose a threat without being in any danger themselves. As abilities are often bursty over sustained - you also don't need to expose yourself to a danger zone for much time. Even with medium/short range mage like Annie - you skip up to a ~600 distance spot, unload everything while incapacitating your foes, then immediately back off before they can respond.
For most AP champions - mana sustain allows them to be safe enough to bypass needing HP sustain.
Once team-fights happen and it would be nice to sustain through a fight so you can deal sustained damages, AP champions have Zhonya's Hourglass for a burst of survivability so you can burst in fights, and with enough CDR occasionally burst again.
It's also worth noting hear that the spell vamp is available in larger quantities than life steal through items.
, and even AP autoattackers like 
, so why not let AD be even with AP people? Is that your point?