I agree that this is a champ by champ notion, and as for Zyra I never found that much trouble against her even though I take
support. (Wierd huh?)
But my post was meant to be a generalistic guide to why these mages work the way they do.
For example, take two of the common mages being played support right now.
and 
Their CC is next to nothing. They bring mostly damage. They like having gold and levels so they are gimped. But they have strong bases so they are viable bot.
Late Game their performance will vary wildly, but they overall provide nothing but damage. That's okay, sometimes team comps don't need as much utility.
But then we have mages like
and 
Can you imagine the chaos that would ensue if they had the base damages of a midlaner (like Brand and Vel) but also had the utility they currently sport now? Why would I take Brand mid if Janna could do comparable damage AND be ungankable?
You see, you are correct in saying that some support mages can be played mid with tweaks, but if you notice the ones you mentioned have little utility, in this case CC.
Sona has little to no CC and is squishy as tissue paper, so if she was changed to be a viable midlaner and support this might work out fine.
However a Support mage like say, Zyra with her snares and plants and all that, would destroy other midlaners becuase She has the Excellent Base Damage of a Support but also has the scaling and CC that beats out other midlanerss
What I can see here is that mages fall into a spectrum of having Utility--------------------------Damage.
The ones more on the damage side can work support IF their bases are good enough, but they won't crowd out other choices because they are mostly meant to cheese kills and get snowballed and in the midlane they are in their natural habitat.
However mages more on the ultility side have to be watched to make sure that they aren't too good in a solo lane becuase they will have the damage of a midlaning mage but have the safety and peel of a support mage.
Make sense?