No, I would say it is not. The word "with" in this case is not interchangeable with "involving in any fashion", it means using that weapon to make a melee weapon attack, but one that doesn't necessarily have to be the Attack action.

The question is really asking, to my mind, is SWS a gish spell? Well, let's take a look at some of the classic 5e gish spells, ones we know have interactions with things like GWF and GWM:

Booming Blade

As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range, otherwise the spell fails. (SCAG p.143)

Green-Flame Blade uses the same language.

We could even take a look at a Xanathar's spell, though it's drastically different:

Shadow Blade

You weave together threads of shadow to create a sword of solidified gloom in your hand. This magic sword lasts until the spell ends. It counts as a simple melee weapon with which you are proficient. It deals 2d8 psychic damage on a hit and has the finesse, light, and thrown properties (range 20/60). (XGtE p.164)

Now, none of these things really sound very much like "You flourish the weapon used in the casting and then..." To my mind, this would suggest that 'flourish' and 'weapon used in the casting' language is much more of a flavor cue and not meant to suggest that the spell will have interactions with GWF and GWM any more than any other melee spell attack would, whether or not you're casting a melee attack spell while holding a greatsword.

No, you are not within 5ft and so do not have advantage

The spell states you 'vanish' (whatever that means mechanically), make the attacks, and then after all the attacks are done you can teleport to various spaces. Nothing says you teleport to each target to make the attack. Likewise, you would have disadvantage to hit prone targets.

I agree the flavor of the spell is that you are moving so fast you can run up and hit all the targets, but that is not what the spell says it does. This is similar to how you don't actually use the weapon you are holding to make the attacks. Additionally, don't let the fact that you are making melee spell attacks throw you. You can make the 'melee' attack because the spell says you can, regardless of distance, and even a regular melee attack wouldn't necessarily have advantage vs a prone target (if you were using a reach weapon for instance).