Rules as Written, No
Flame Blade has a specific action that uses a melee spell attack. It is not an attack action. An attack of opportunity only requires a melee attack, not the attack action, so that does not immediately disqualify it.
What disqualifies it is that the spell specifically requires an action to use (a reaction is not an action). The blade is not literally a weapon, and so it cant be used for multi-attack or opportunity attacks. It specifically requires an action to make a melee spell attack. The reasoning might be that it requires some mental ability to concentrate the blade or to hold on to, but you're not using your physicality to wield it. Whatever the reasoning, it was not designed to be used exactly as a weapon is used.
Jeremy Crawford also agrees
The intent is no.
It should be noted that you can make opportunity attacks with other weapons, provided you meet the circumstances for that to occur. Wielding the flame blade does not prevent you from making AoO, you just can't make them with the flame blade.
Obviously, if your DM allows it, there's no logical in-world reason why you couldn't take a swing with it. However, 3d6 fire damage is a pretty powerful opportunity attack.
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.
Best Answer
No, you can't
The rub is that it describes it as a melee spell attack, vs. a melee attack. As such its in the same category as Inflict Wounds and other touch spells as a distinct action vs. a regular Attack action.
Extra Attack only triggers when you make the Attack action
Even if you interpret melee spell attacks as being one and the same as melee attacks, the simple fact remains that Extra Attack only triggers when the Attack action is taken.