The spell is garbage. That is the main problem here. The upcast scaling is terrible. Even if you are a level 20 fighter who somehow gets access to this spell at 9th level, we are speaking of only 6d6 damage per hit (without any stat modifier) if cast with a level 8 spell. It still needs to hit (without being a magic weapon) It is single target and takes your concentration slot.
So we are speaking of 24d6 damage per round (48d6 damage with action surge) in the best case. For an 8th level spell. For a lvl20 fighter. Ok. I admit. That is not that bad. But that is nigh impossible with normal rules. The best you can achieve is level 11 fighter eldritch knight with 9 levels of druid and someone else casting haste on you. You have great con saves and 3 attacks (4 with haste), action surge and level 6 slots.
So it is 15d6 damage (30d6 with action surge), 20d6 with haste (35 with haste and action surge).
That averages only 122.5 * hit chance damage in the first round of combat. Costing two concentration slots, a bonus action to cast from you. An action from the other spellcaster. A level 8 and a level 3 spell slot and you run the risk of losing concentration or worse the other spellcaster losing concentration. After that it averages to just 70 * hut chance damage per round.
Meanwhile a great weapon battle master level 20 with a great sword, precision attack has 4 attacks of 2d6+15 damage and precision attack to negate the -5 to hit. And a simple level 1 spell like bless will get you another bonus to hit, and cast with a level 3 spell, probably the whole party will benefit too (including the spellcastter boosting thwir own concentration saves). Not to mention magic weapons that are probably available at that level.
So in the worst case we are speaking of 8d6+60 damage. For the first 2 rounds 16d6+120 damage (two action surges), even leaving the bonus action free for a probable 9th attack from great weapon mastery. That averages 162 * hit chance damage for the first 2 rounds and 88*hit chance therafter (maybe a little bit reduced hit chance due to having no superiority dice left.. but since you started well ahead the flame blade user, no issues here).
I am really wondering how anyone in their right mind considers this spell broken in any way, even when used woth a high level spell from someone with extra attacks.
I think the best fix would indeed be just improving the scaling to 1d6 extra damage per upcast spell slot and maybe just removing concentration. The spell is not worth using otherwise.
10d6 fire damage with a melee spell attack at level 9 does not sound broken, does it?
Edit: the new true strike or shilelagh (old or new) easily surpass flame blade. I'd take a cantrip for 1d8+spell casting modifier without concentration over it 3d6 for a level 2 spell slot and concentration every day.
I'd use the spell slot and concentration for fairy fire and happily whack my opponents. Or I just cast flaming sphere to add 2d6 fire damage to possibly more than one opponent as a bonus action and also having some area controll.