The objective of saying "aaa" while the doctor examines the throat is that he can see the pharynx and check if the mucosa is inflamed , or has pus. Normally the tongue does not allow a good view of the pharynx, but when saying "aaa" the back flattens , the tongue extends into the hollow of the lower jaw, and that allows the throat to be seen.
Other vowels would not be suitable for this purpose. For example, "iii" is pronounced too forward. And "uuu", being a back vowel for whose articulation the back of the tongue is raised towards the soft palate, further reduces the visibility of the throat.
