The large general aura provides a huge, benefit to ability spamming, mana-using champions. It makes a number of different champions viable and able to repeatedly fight to take and then defend a point against attacks rather than going through a full fight and then having to back right away because of low mana, (i.e. has some mana left but not enough for an extended fight as tends to happen in the 1v1, 2v2, scenarios). I'm thinking of champions like: Talon, Warwick, Blitzcrank, Skarner, Jarvin, Jax, Pantheon, Sion, Trundle, Udyr, and Xin, just off the top of my head. In the early game a couple full rounds of their abilities and you are usally about out of mana with SR or TT mana regen. I fell like taking the mana regen away from these champs and giving it just to mages would really hold them back and drive them down to lower tiers.
Another thought, if you took away the generalized and gave specific buffs to specific champions or classes what would that do for some of the unusual but viable builds? Examples, Master Yi, do you give him the damage buff or the mana regen and magic pen buff? Same question with Sion and Shaco? Other even less common ones include, AP rengar, AD Kennen, Teemo (AP, AD, On-hit hybrid), Alistar (AP or AD), Malphite, Kayle, and I'm sure others that I'm not thinking of, what specific buff do you give them?
An idea, if you wanted to narrow the buff down what if you broke it down into various different buffs like, Armor pen, Magic pen, Attack speed, Mana regen, (maybe others) and let the player pick at the beginning which 2 buffs they want. This would allow the player to customize even further the style of play they want to have (kind of like Kha Zix who you can customize with both your build and your evolutions).
TLDR: Taking away mana-regen buff would hurt a lot of champions early game imo, only giving certain buff to certain classes would hinder fun off builds. I'm sure there are work arounds that could be done by riot (you guys do this for a living but that's my opinion.