Is there any reason I couldn't just play a bard with a bow and use the Blade flourishes?
I'm pretty sure I missed a word somewhere, because after reading through the whole thing, I came to the conclusion that the only time the word "melee" came up was in the fighting styles and the use of a melee weapon for a focus (the latter of which doesn't matter with bows, since you can free a hand to cast).
Some of the effects aren't valuable to a range-specific character, but I'm strictly talking semantics.
Best Answer
Ranged Works!
As you surmise, the only requirement for the Blade Flourish is (my emphasis):
If you've taken the Attack action, then you can Flourish. There is no requirement for Melee weapons, so Ranged works just as well
Ranged weapons for Bards
Do note that a single-classed Bard is only proficient in the following ranged weapons: