I'd just like to point out that, by treating a weapon as an animate object except for its HP, flying is an altogether terrible property to dump on an otherwise strong item.
As far as I can tell, animated objects only get a slam attack based on their size. Rug-like objects can blind and rope-like objects can constrict, but as far as I can tell there's no special attack for blunt or bladed objects allowing them to make weapon attacks.
Any reasonable DM is going to say a +1 flaming flying great sword is going to deal 2d6+1 slashing + 1d6 fire, but RAW as an animated object it'd just gets a 1d4 slam attack.