Excuse me if I seem a little critical about this, but I don't think Brand needs a rework. I think he's in a decent spot at the moment. As for the suggested rework ideas, it maintains Brand's identity as a generic fire mage, but it heavily infringes on everything else.
If Brand needed a rework, I'm not sure if this is the right direction. With the ultimate change suggestion, it puts majority of his kill potential on his R since he has access to all three of his upgraded abilities. This means, in terms of balance, all of his basic and ultimate upgraded skills need to be nerfed/changed heavily to compensate the huge powerspike at level 6. Also by doing this, it effectively nerfs his early game potential as a lane bully, which is required for him to scale relatively well into the mid-game.
Additionally, by suggesting changing his Passive and his E to new alterations, which also probably means getting rid of all the additional ability interactions with Blaze, this will severely cripple his kit, utility, and his playstyle. Without Blaze he cannot effectively wither down his opponent (especially against tanks) and be as effective in teamfights.
By removing Blaze and its additional effects, it removes the depth from Brand's playstyle and makes his kit incredibly linear. The suggested change will make his skills completely reliant on the damage from spells rather than their utility. The Player will no longer need to consider the additional effects of skills or the rotation of skill order, but rather he or she would be spamming spells for the sake of damage. To me, this is pretty unfun to play.
Without the reliability of his E, the Blaze effect on said E, and the stun from Q, Brand cannot efficiently combo into his E>Q>W. This will tremendously affect his trading as well as his damage output. From the description of the suggested E, it will become a lesser replica of Morgana's W. Although it would be somewhat effective as a harass tool, it is incredibly inefficient due to it being both easily avoidable and a DoT. As a DoT, its damage would be lowered substantially and become inconsistent. On the other hand, Morgana can utilize the full effectiveness of her W since she has a Snare to capitilize on the DoT. However, the suggested Brand change doesn't have a stun or slow, thereby, the DoT's effectiveness is significantly lowered.
**TLDR; ** I think this rework suggestion needs a little more work. When reworking a champion, you want to hopefully achieve the pinnacle of its thematic persona as well as maintaining/upgrading the champion's gameplay. As of right now, to me, it seems to be going in the opposite direction.