Two things to consider:
Firstly, when determining the size of a circular or spherical object, we would usually use the diameter, so the straight "chord length" is what you are trying to measure, not a slightly curved arc which deflects behind the straight chord.
Secondly, for small angles, such as astronomical targets, the straight chord length and curved arc length are almost identical anyway. The curved arc length is equal to the angle (in radians) times the radius, while the straight chord length is equal to 2 times the sine of half the angle (almost exactly equal to the sine of the angle, for all practical purposes).
Consider the Moon and the Sun, which are bigger than most targets, apart from the biggest galaxies and clusters. They are half a degree across; the curved arc length works out to be 0.00872665 times the radius, while the straight arc length is 0.00872662 times the radius, roughly 3 parts in a million difference. For smaller targets, the difference is even smaller.
