Current situation
Flame Blade is a 2nd level spell, and a very weak one:
You can use your action to make a melee spell attack with the fiery blade. On a hit, the target takes 3d6 fire damage.
This is on average 10.5 damage, while Shillelagh does 7.5 with 16 Wis, with 20 Wis it is 9.5.
So in exchange of 1 damage, Shillelagh does not need Concentration, and its damage type is also more beneficial. A lot more creatures resist Fire than are vulnerable to it, but magical bludgeoning can affect everyone.
Comparison to Flaming Sphere
Flaming Sphere does the same type of damage and also needs Concentration, but it is usable with a bonus action, has the chance of damaging multiple enemies, and does half damage even if the enemy saves successfully. If we assume (as the DMG does) that attacks can easily be converted to saves, and take a 50% hit chance, both spells have the same damage output.
Adding Cantrips
While sustaining Flaming Sphere, you can use a cantrip with your action, giving further advantage to it.
So for the same investment (same spell slot, concentration, all your available actions) you can do much higher damage with the combination of a cantrip + Flaming Sphere.
Adding Spells
While concentrating on one of these spells, you see Treant approaching. You cast Blight with your action. If you used Flaming Sphere you did 8d8+2d6 damage this turn, but only 8d8 with Flame Blade.
Proposal
The initial damage is 4d6 and it increases with 1d6 per spell level, instead of every second one. This is still weaker than Shillelagh + Flaming Sphere, but at least now it has some use cases.
FB2: Flame Blade with 2nd level slot
FBm: Flame Blade with the highest available spell slot
Sh + FSm: Shillelagh as the action, and Flaming Spehere with the highest available spell slot as the bonus action
So the questions:
1) Did I miss something, can this get overpowered in some situations?
2) Would another 1d6 be overpowered? It is still less than Flaming Sphere + Produce Flame after level 11
Best Answer
Flame blade has a duration of concentration, up to 10 minutes, while flaming sphere has a duration of concentration, up to 1 minute and shillelagh a duration of one minute (no concentration).
Flame blade provides 3d6 magical fire damage using your Wisdom modifier (melee spell attack) for 10 minutes. This could potentially span two or three combats (likely, if you're in a tightly packed dungeon), and allows you to use other spells or cantrips as you please as long as they don't require concentration.
Flaming sphere, on the other hand, typically lasts no longer than one combat (10 rounds at most), and uses the same level spell slot as flame blade.
We can do the math and see that economically, when the hit ratio is the same (as you have pointed out in your question), flaming sphere does only 1d6 avg "guaranteed" damage, and can only do that 10 times. I say "guaranteed" because it only does damage when it rams an enemy or when an enemy stops within 5 feet of it, and the enemy still gets a save against it. You can control when it rams an enemy, you can't control when an enemy stops near it.
On the other hand, flame blade can hit for 3d6 (that's an additional 4 average damage over flaming sphere if the target fails its save), and it can do it 10 times as many times. At an average of 11 damage and assuming a perfect hit ratio and attacking every round, that's 1100 potential damage over 10 minutes. Flaming sphere, on the other hand, can do 70 at most (and under perfect conditions, with everything going just right, 560 total avg damage).
Finally, flame blade has utility beyond being a weapon: it also produces light that can be dismissed at will and evoked using a bonus action while the spell is still within its casting duration. This is more useful than the light provided by flaming sphere, which in this case burns twice as bright but only a tenth as long.