Yochi, I don’t know you so it’s not personal, but why would you think that you would know what you missed? I would think that it may be the ones that went, and got something from there, who would possibly see what you can’t.
The question is too broad. It certainly depends on the individual and depends on where they go. For example, one who learns in Brisk and is interested in that, and is also from the true mevakshim that are looking to really get from what authentic Brisk has to offer, can get something there that they can’t get anywhere else in the world. Additionally, America is driven by materialism which in most cases waters down the purity of our yiddishkeit. Again, depending on who you are and where you go you can spend time associating with and being inspired by a type of yiddishkeit that is mostly not available in the USA. Where one lives simply, eschewing materialism, focused on spiritual growth and living life entirely through yiddishkeit, values uninfluenced by secularism, etc. It can be a real eye opener, and at times life altering.
Additionally, there are great tzaddikim and talmidei chachomim that one can grow from by association.
Girls too, depending on their upbringing, background and where they are holding in life can grow immeasurably from the above.
Granted, many, some would suggest most, do not maximize the opportunity and some would posture that the overall benefit do not justify the risks, from a communal position. But, the OP asked what the purpose is. On an individual basis this may be some ideas.
Lastly, as far as boys are concerned, once a boy is past 4th year b”m he does not really have any serious local options. Just about everyone goes to EY. Furthermore, when he comes back to Lakewood and needs chaburos, chavrusos, etc, depending on where he learnt previously in America, his friends and yeshiva mates from EY may be the difference in a successful experience in BMG or not.
Granted, if we are discussing the system we can discuss creating opportunities locally, but until such a point in time the facts are that for most there is really no other option