It’s a sensitive issue, but, for me at least, there are differences in methodology between selecting one’s personal rav or posek and formulating an opinion about more public figures with international notoriety.
Choosing one’s rav is more about compatibility, approachability and accessibility than knowledge, provided the rav has the integrity to recognize his own limitations. Issues beyond the knowledge and experience of any rav can be addressed in consultation with other rabbanim possessing greater knowledge and experience.
However, as far as public figures are concerned, the best approach is simply to listen/read pronouncements made in their name and then refrain from public criticism of those pronouncements or the person even on topics one feels strongly about. There’s just no way anyone can judge from a distance the character and abilities of those publicized as Torah leaders.
In what I think are the vast majority of times, people choose and then root either for or against a particular gadol the same way people choose to root for against sports teams or athletes, l’havdil. It becomes more about one’s personal pride (he’s the rav I chose or he supports *my* opinion on the matter, so I proclaim his merits or, r”l, the opposite) than an honest evaluation of the issue or person.