So the thing with Flame Blade is- it isn't a weapon. The spell description says it has the shape of a scimitar, but it never states that it is a scimitar. Why does this matter?
Because you can use Flame Blade without needing any particular weapon proficiency. You might say that this is an oversight because the spell isn't found on the list of a caster who can't use a scimitar, but imagine you have a Wizard with a Ring of Spell Storing and the party Druid put a Flame Blade in there.
They could "cast" Flame Blade from the Ring and use it just fine without needing any proficiency- thus Flame Blade isn't a weapon.
Because it's not a weapon, it can't benefit from Fighting Styles or Sneak Attack, or any other special ability of class that primarily uses weapons. I'm fairly confident it was never intended to be used with Extra Attack either (since this is something Druids, the class that is intended to primarily use this spell, lacks).
Now because it's not a weapon, you could allow it to be used as an improvised weapon...except it's a spell effect, not an actual object. Probably. The spell doesn't say it's an object, so it comes down to how you feel about "fiery blade".
If you do think it's an object, then you could allow it to be used as an Improvised Weapon, but this leads to some things you need to consider, the first of which is what damage does an improvised Flame Blade do anyways?
Is it now the equivalent of a torch? You might say this doesn't make sense for it to be less effective, but that's pretty much how Improvised Weapons function- they are shoddy replacements for the real thing, mostly used in desperation.
If you do decide that the "improvised" Flame Blade deals the full damage when used in this way, since, after all, you lose your proficiency bonus, then you have to ask yourself, what about Tavern Brawler?
Do you feel that a character can take a spell intended to be used by one class in one way, and build around it to make it vastly more effective when not being used by a single-classed Druid? Or even by a Druid at all?
This is the primary point of contention with character optimization- is it right to try to derive more benefit out of something than it was meant to provide? This path leads to Life Clerics using Goodberry.
Personally, if someone wants to bend over backwards to get more benefit out of something, I'm generally ok with it, but in this case, I don't particularly think the juice is worth the squeeze. Sure, a Fighter built to use Flame Blade could get a lot of damage out of it, but is it really the best thing they could be doing? Especially when you start to encounter enemies resistant/immune to fire damage regularly as the game progresses?
One fire elemental and the player of the tricked-out Fighter has hamstrung himself and will be glaring at you, lol.