No one can know why someone becomes a Gadol. Rav Moshe was asked by Time magazine how he came to be considered such an important figure in Orthodox Jewry and he answered that it just evolved that way. One Shaila followed another and one Psak followed another and was accepted. In other words, the whole thing just snowballed. Each successive Shaila and acceptance of his Psak added to his stature until he achieved the status as one of the Gedolei HaDor. (...and in the eyes of many THE Gadol HaDor.)
It is a widely held view that the way one can achieve Gadlus is by taking advantage of the system. That is, to send one’s sons to the finest Yeshivos with the best Rabbeim. As R. Eliyahu Dessler points out, you throw 1000 young men into the Beis HaMedrash and hopefully one will rise to become a Gadol.
But I submit that this is an unlikely way to do it. Conformity does not produce an individual that is so unique in his Torah learning, knowledge, and any other requirements... that he will become a Gadol. To be sure (I hate that phrase) any individual who attends an institution like Lakewood, if he is diligent, will become a major Talmid Chacham and perhaps even a Posek. But a Gadol? I’m not so sure.
Most often there are factors extant in an individual or circumstances surrounding him that are not obvious to the casual observer. If one looks at it historically, most Gedolim do not follow the normative educational path. Often they are prodigies given special treatment. Such people are unique individuals who would've come to Gadlus no matter where they went to school. They did not chase Gadlus. Gadlus found them.
Bearing that in mind I would say that experience outside the "normal" environment of a Charedi Yeshiva is a plus in the development of Gadlus. It enables Gedolim to have perspectives beyond the narrow confines of a Charedi Beis HaMedrash. A Gadol who attended a Yeshiva with distractions such as sports, movies, TV, and a myriad of other such distractions yet finishes all of Shas (for example) with all of those distractions surrounding him has achieved a great deal more than someone who has finished Shas without any distractions at all in a Lakewood type environment.
This is not to say that all those distractions should be required. Of course not. But in those circumstances where they exist, if one “overcomes” them, he is a far greater individual than someone who had no distractions at all to worry about.
An individual so exposed becomes better equipped to handle these and other distractions in his future.
But it is so much more than that. He has a much broader world view and understanding of it and is better equipped to handle issues of the day. His secular knowledge is initially broader, which will help in areas of Psak Halacha that require such knowledge. He knows the “mind” of other Hashkafos better than those who only attend a Lakewood type institution. In my view a Gadol with more exposure than is typical for a Charedi Yeshiva student will be far greater than one who has limited or no exposure to anything other than the Daled Amos of his own Beis HaMedrash.
Of course one might say that Rav Moshe and most other Gedolim weren’t exposed to formal study of secular knowledge, other Hashkafos, or distractions of the type I mentioned above... during their own development. Are they lesser Gedolim because of it? No, of course not. They were far greater than any other individuals in our own time. But that was then.
What I am saying is that in our time it is necessary. Technological advances have exploded to proportions unimaginable even a generation ago. Distractions of every kind in far greater number and variety exist that are too easily accessible. The more you know of both general knowledge and about the cultural environment of your own milieu, the more prepared you are to achieve the great heights required of a Gadol. And perhaps this is one reason that the cookie cutter system of our Yeshivos has not yet produced a Gadol of the stature of Rav Moshe.