Rapiers aren't as heavy or clunky as you're probably thinking. I mean, yes it's heavier than modern sport blades, or the small sword which gained popularity in the 18th century, but as a thrusting weapon, it's still lighter than something like a cavalry sabre. I have twigs instead of biceps, and can fence rapier adequately. The force from an attack should come from your legs, not your arm, so the only time you need to hold a rapier straight is for the split second before a lunge.

The point of balance is only slightly forward of the guard. You can control the tip of a rapier quite capably with just an index finger and thumb.

"Distal taper" is when a blade narrows towards the tip, and it accounts for significant weight savings when compared to a blade of uniform thickness.

But by the 19th century most duels were being fought with pistols. You'd maybe have military officers duking it out with sabres, and the surviving French nobility might still have had a soft spot for smallswords, but firearms were considered more "egalitarian" and representative of the "democratic spirit" sweeping Europe.