“Other than the fact that it looks cool, what is the practical advantage of a flying saucer design exactly?”
I think the biggest possible advantage would be maneuverability, since you can redirect thrust quickly in many directions. Also, a disc design could remain pretty level moving in any direction, so that could have some uses too I guess.
Or...how about putting a rotating ring around it with a gun mounted on it, so you could target 360 degrees around the ship?
Okay, but how does it achieve lift? Anything motorized that hovers either needs rotors or jets. Surely the whole vehicle does not rotate, only some inner component.
The Russians experimented with a circular battlship design in the 19th century. It was a complete failure; the worst aspect of the design was that off-axis recoil from the ship's guns set the whole thing spinning. I can't imagine that would be any better on a flying gunship than a floating one.