I think its more one big simple issue: Whatever a Fighter/Tank can provide, a Mage can provide better?
So, what would you usually be picking a Fighter or Tank over Mage for? Tankiness and durability. And right now in the game the difference between a Mage dying and a Fighter/Tank dying is negligable. Fighters and Tanks, at the front line of a fight, usually last about 4-6 seconds. They are glorified meatshields at best, throwing out their abilities and hoping to the Rito gods that they don't die. The only Fighters that are worth using are Fighters that deal so much damage and have so much mobility that they almost act closer to Assassins than actual Fighters.
Meanwhile Mages not only have access to more defensive oriented item cool downs, but dont even need to be in "Danger Range" to interact in a fight. Range is incredibly powerful in League of Legends, as seen whenever a ranged champion becomes dominant in Top Lane. They usually just dominate and it feels miserable to play against. And unlike Dota 2, there are no turn rates to make kiting tactics weaker in League, PLUS most of the passive movement speed itemization is given to ADCs and even Mages. So ranged champions not only have more defensive cool down items to deal with melee champions (Fighters and Tanks) but also more passive mobility to help deal with them. Lets compare a list between the roles (and of course some hybrid champions could pull from both, like Ekko, Vladimir, Rumble, etc.)
Mage Defensive Items-

ADC Defensive Items-

Fighter Defensive Items-

Tank Defensive Items-

Assassin Defensive Item- 
Of course, there is much overlap if you are an AD Fighter, or maybe a CQC ADC, or AP Fighter, or such. However, notice the difference between the ADC/Mage Defensive Items and the Fighter/Tank Defensive Items. The Fighter and Tank defensive items are mostly passive benefits rather than powerful limited time effects. Guardian's Angel and Zhonya's Hourglass are two incredibly powerful items that have been rebalanced constantly due to their power. Banshee's Veil and Phantom Dancer are also two not quite as strong, but rather powerful defensive items that are often underestimated in impact. People like to diss on Banshee's Veil, but you will love the item when someone like Zed or Zoe try to kill you in one combo. Meanwhile Fighter and Tank items have had more passive and underwhelming benefits that do much less than provide absolute mitigation. Even worse, most Tank and Fighter defensive items get entirely negated usually by True Damage and other anti-defense mechanics, like Grievous Wounds and such. Meanwhile there is little you can do when a Mage activates Zhonya's or has a spell shield, or an ADC simply has a free partial revive or a shield that passively activates.
And most importantly and once again, this is all compounded by the difference in WHERE these two different groups of classes are in fights. Mages and ADCs have at least comparable defensive items to Fighters and Tanks all while being miles further away from the frontline than both Fighters and Tanks. Mages have been in the highlight as of late because they provide crowd control equivalent to that of Tanks, but are supposed to be weaker due to decreased defenses and usually less mobility. However, as said, thats just not the case due to how quickly Fighters and Tanks die and the ignorable difference in the defensive items of Mages vs Fighters/Tanks.
TL;DR Mages are dominant because their main weakness when compared to Fighters/Tanks are null and void in this day and age. You pick Tanks and Fighters due to how much more durable they are, but thats negated when they die extremely quickly in fights nevertheless.