Brand feels underwhelming to play when considering his intended fantasy.
It's kinda like comparing the McDonalds commercial (THE LIVING INFERNO!!!) with the actual product (generic fire mage #324524).
And I think I have located the primary problem: Thematically Brand is supposed to be most effective at close range. He's not the guy throwing fireballs. He IS the fireball. At least in terms of lore. Yet that doesn't deliver for some reason.
Here are some ideas I'd propose: Brands abilities interact with him. Being in the area of W heals and restores a bit of mana, E can now be self-cast for the AoE effect and a short damage amp (FIRED UP!) and the ultimate can now bounce of Brand, also, he can charge his passive on himself, but it doesn't damage him.
Of course that would have to be balanced, but I think this is how it could work without causing too many issues. And before anyone says that the ult would be OP, then I'd say that it would be still weaker than what other mages have to offer: Annie, Veigar, etc. so it might not be too much of a deal.