Why does area get to be especially fun and definite while length, its one-dimension-away sibling doesn’t?

Excellent question, and as you yourself allude to, it’s a question of bounds. If you can establish and upper and lower bound on a quantity and make them approach eachother, you can measure it.

On a finite 2d surface you can make absolute lower and upper bounds on any area - lower is zero, upper is the full surface. All areas are measurable. But on the same surface you can make a line infinitely squiggly and detailed, essentially drawing a fractal. So the upper bound on the length of a line is infinite. Which means not all lines have a measurably length.

This extends naturally to higher dimensions - in a finite 3d space, volumes must be finite, but both lines and areas can be fractally complex and infinite. And so on.