When you hover the mouse near the center, the center point is used as reference. Hover it near one of the ends and the end will be used as a ref point. Doing it that way, it is on-grid
But to clarify, I think you are referring to the mid-point (halfway point) of the arc and not the radius center point. (??) Any of those four points seem to be usable as “handles”. Interpreting your response to mean the arc halfway point, what you say appears to be correct. I have gone back and tried it.
With a 0.5 mm grid, a 90 degree arc, and a 2 mm radius, the radius center point and the two end points should all be on grid. But the arc halfway point is not; there is a sine and cosine of 45 degrees involved.