Right, but this makes the area of the cover twice the area of the hole, and the cover twice as heavy as it would need to be if it were round. As Feynman pointed out, even round covers are already pretty heavy—I injured my wrist (ligaments?) from the tension of lifting one in my teens.
However, if this were the only consideration, you could get a wider-diameter manhole with a smaller cover by making it in the form of an equilateral triangle, as evidently Nashua, NH, does: https://en.wikipedia.org/wiki/Manhole_cover#Other. Indeed, you don't need to stop there; if you curve the sides of the triangle in toward the center, you can get a manhole with an even smaller surface area for a given diameter, although at some point the extra "diameter" will be too curved to be useful.
Indeed, though to do it this way would be very wasteful of resources as the ratio of the manhole size to the size of the cover wouldn't be maximized. The cover would be needlessly covering a lip whose large size only exists because of the poor choice of shape.
However, if this were the only consideration, you could get a wider-diameter manhole with a smaller cover by making it in the form of an equilateral triangle, as evidently Nashua, NH, does: https://en.wikipedia.org/wiki/Manhole_cover#Other. Indeed, you don't need to stop there; if you curve the sides of the triangle in toward the center, you can get a manhole with an even smaller surface area for a given diameter, although at some point the extra "diameter" will be too curved to be useful.