As a Ahri main, I am not a Ahri Board person, whatever the heck that is. I am just an analytical person who tries to give the best responses possible on whatever topic.
It is the open variety that classes have and is something mages lack. They MUST go AP and cannot waste a slot on anything else.
Marksmen have options: Attack speed options, Critical strike options, AD options and life steal options. (Crit is never without Attack speed or AD).
Bruisers have options: AD options and survival options.
Tanks have options: HP options, armor options, and magic Res options
Assassins have options: Some are AP while others are AD, survival options.
Supports~~ do not~~ have options: Sightstone, survival, ally oriented.
Mages have options: AP options.
I know you are going to say that mages can get armor or magic res but I mean they need something that is meaningful. Irelia, Illoai and Nasus can go for full AD/AP or some AD/AP with HP.
Lucian, Sivir, MF, Caitlyn can go all AD, All attack speed, as much crit as possible, as much life steal as possible or any combination of them.
Fizz, Kat, Zed, Yasuo (if we think of him as an assassin) can go full AP/AD, get some mr, get G.A.,
Sion, Sejuani, Rammus can go all armor, all magic res (so they have AD by defualt), they can go AD/AP if they want to, and go all the HP
Xerath, Cass, Ahir can go AP.
Mages can't stray from that path.
The idea I am trying to get across is that if you were to see a most commonly used items with each sub-class, most items (not including boots) would have lower percentages meaning that the classes are flexible and adaptable, with bruisers and tanks. And with marksmen or assassins, they have options. Mages do not. They have only a few AP items and some of them are for support like Ardent Censor or Frost Queen's Reclaim. Excluding those items, options are not open.
You want survivalilty as a mage, you got two options: Zhyona's and Abyssal Scepter- armour / magic res. You want HP? You got Rylai's, Proto Belt, Liandra's, GLP-800 and Rod of Ages. Okay, 5 options. Not a lot of AP, mr nor armour but options. Then it is the matter of the passive of them. Getting both Proto belt and GLP-800 is pointless because shared cooldowns. Rod takes time so that is a early game choice. Liandra's good against tanks. Rylai's if you really need the cc. Plus, with the play style that mages are suppose to be in the back like, there is no need to get that much HP.
It is 11 at night here and I am tired so I will have to call my one-sided, no counter argument addressing paper in and post it... unfortunately.