I had a similar objection and idea for alternative metric (with area and perimeter). But for most countries, the perimeter is ill-defined. See: https://en.wikipedia.org/wiki/Coastline_paradox
There does not exist an adequate methodology to determine the length of a nation's coastline. Even if there did exist such a methodology, the various surveying bodies of the world's governments do not agree on any methodology. So not only are the actual lengths of a country's coastlines undefined, we don't even agree on the least bad way to calculate it.
Areas are relatively reliable despite the fact that the methodology can be wildly different between two surveying agencies. Area errors tend to cancel out. You have a little more land here, a little less land there. But perimeter error nearly always increases the perimeter, and rarely ever decreases it.
The dataset defines the countries as polygons. The perimeter of a polygon is well-defined.
Your argument that perimeter error nearly always increases the perimeter seems completely wrong to me. In a perfect snapshot of a coastline, the perimeter would be fractalized at the size of a grain of sand and would define a maximum perimeter. Any approximations would tend to smooth this out and reduce the perimeter.
But regardless of whether we can make a perfect measurement of a coastline, for the purposes of this calculation, we're looking at the country represented as a polygon on the surface of a sphere. Both the area and perimeter of such a polygon are well-defined quantities. If it makes you feel better, we can say that we're not determining the roundness of the country but the roundness of the representation of the country in the dataset (which is what the original story was also doing).
Go look at Iceland on a map. Don't just imagine it in your head, actually look at it. Now look at a map of Jordan. Ask yourself which of these countries is more circular, and then ask yourself which of these countries your metric will tell you is more circular.