"The "dream" is to immerse in learning while being surrounded by a critical mass of intelligent, driven, and most importantly well-connected people. There's a reason a large majority of the top entrepreneurs seem to have come from the same set of schools."

One thing I am having a hard time grasping with your description of your desire is, what is the end goal? Do you want to learn amazing things about an academic area that you really enjoy? Do you want to meet well connected people? Do you want to be a fantastic top entrepreneur? Because these are all pretty mutually exclusive. I did not go to a top university but I do know well connected people and I am a relatively successful entrepreneur (not a top entrepreneur by any means). I will try to give my background as an example.

As I youth, due in large part to my fathers mathematics background, I gained a love of theoretical mathematics. I was top of my class in high school and graduated early. I went to college and got scared about an academic career, due again mostly to my father who was an entrepreneur and engineer. So I switched majors a few times and landed on law. I dropped out of law school to be an entrepreneur and engineer myself, with the ultimate goal to support my rapidly growing family.

By worldly standards I was raised relatively well off in Southern California but I am no Bezos, Graham, nor Page. I went to state schools and have a non-technical degree. I have started and been successful in two companies, work for myself as a consultant now, and all things considered do exceptionally well. But a few years ago my goal of supporting my family became easier (success in business helps a lot) I got the itch to get back into academia. I have been doing "amateur" theoretical mathematics research for fun for years, and I haven't felt fulfilled in business or engineering in quite a while. So I came up with a plan.

Step one, while I would love to go to MIT or Cal Tech, I don't see that ever happening. So, go back to a state school and become very familiar with the staff there. I found that most of them DID go to those prestigious schools and studied under phenomenal professors and have amazing knowledge and interesting theoretical mathematician and physics. They also have amazing contacts to share. So I am working on receiving a bachelors in mathematics, then a masters, then a PhD from those folks, and really enjoy the experience.

Second step (concurrent to first), continue to do well in my business areas (mostly B2B application development consulting as well as some recurring revenue products). Will I get Graham/Bezos/Gates rich? Heck no! But I can get enough to make my third step and ultimate goal possible.

Step three, through the local university fund my own theoretical mathematics research. Why? Because I really enjoy mathematics research. And funding for theoretical mathematics research is REALLY hard to obtain, there isn't much of it, and there are some really brilliant minds in the field who deserve it. So, I use step two to fund step three.

The point is, I saw what I wanted and I found a way to get to it. I have very specific steps around each of those steps (step two, for example, is far more complex than I have written out here, and has milestones and subgoals). But it is working and I can pursue my academic dreams while also supporting my family and myself well.

That all being said, I understand my steps might not work specifically for you. As you mention, you need focus for academic discovery. All of our brains work differently and I have been blessed to be able to do these things concurrently. But the more important point is, if you have a goal, make steps and work to get to it. You may find that by making steps to get to your ultimate goal, you have to change that goal slightly (as with me, I will never work at MIT or Cal Tech). But you'll still find the satisfaction of making it, lose the anger and the envy, and ultimately be happier than you are.