Apparently they saw a change in direction of the light. They attributed it to the gravity of the sun.
Could it not be from passing through the atmosphere of the sun caused a refraction which is known to change the direction of light?
I don't have the background to allow me to say what refractive effects are possible in such a situation.
A line drawn from an observer to the observed position of such a distant star would establish the apparent position of that star. I would imagine that the apparent position was considerably outside the disk of the sun and sufficiently outside to establish that such rays did not pass through appreciable solar atmosphere. One would have to know more about such "atmosphere" in order to be confident. I have never heard that such was a consideration so I am inclined to believe that it was not a factor.
I have read that the measurement error in the original observations were such that there was some significant probability of reaching the correct conclusion by accident of measurement error rather than by having actually measured the displacement. I haven't heard any challenge that would suggest that observations made with a more perfect experimental setup would violate the theory.
I believe that the "bending" of light rays by gravity has been separately confirmed by the action of "lensing" by very dense objects in the universe. Objects behind such dense obects can appear distorted because divergent rays become aligned causing the object to appear larger.