That said, based on your answer, we've just defined away the concept.
Correct, which I'm going to say is proof of my concept/thesis that a "self made billionaire" does not exist, and is an inherently flawed, fantastical-thinking concept, if taken literally. However, my second contention is that the notion of "self made billionaire" is really just a euphemism for signifying belief that the person has "earned" or "deserves" the wealth.
If a person became a millionaire by collecting littered cans, there's no way we'd say they didn't deserve to be a millionaire.
Putting aside the fact that you said "millionaire" as opposed to "billionaire", I have to admit that at least philosophically, I'd have to concede that I'd tend to agree with you on that. It's pretty miserly to try and argue that they don't deserve it. However, this was such a lovely thought exercise that I went ahead and did the math. TL;DR, this hypothetical fails because its not actually logistically possible.
So first of all, this hypothetical is immediately rendered impossible because we have to make some baseline assumptions that aren't even possible. For example, our protagonist will work every single day with absolutely no time off for sickness, injury, holidays, family events, etc., and the protagonist will never take more than his 8 hour per day allotment for all his can-cashing-in, eating, sleeping and pooping needs and will work tirelessly for the other 16 hours every single day. Finally, we assume that our naughty protagonist will never ever pay a single dime in taxes and we will give him a pass on that, in terms of being "self made", despite you explicitly referencing that this would be in your view, disqualifying. Finally and most importantly, we will assume that our protagonist is able to somehow obtain food and drink for absolutely free and never has to spend a single penny on his own sustenance... again, we will give him a pass on that in terms of being "self made", or we can just assume he drinks rainwater and eats wildflowers, or garbage or cooks roadkill, whatever, as opposed to panhandling (which he doesn't have time for anyway) or accepting "handouts".
So again, putting the impossibility of the above aside, here is how we get our hero his millionaire status. One million divides neatly into $0.05 returns for cans, giving us 20 million cans needed to be collected to reach the million dollars. Since we aren't going to make our protagonist a truant child, we can start him off on his journey the day he turns 18 and no longer has to attend school or be under his lazy evil wicked stepmothers thumb. We will give him a longer than average life, and let him make it to 78 years old, despite the backbreaking journey he is embarking on. So that gives him 60 years to do his life's collecting work. That works out to 333,333.3(continuing) cans per year. That means he will need to gather 913.24 cans per day, which, based on a 16 hour "Frost Punk" style work day, that works out to roughly 57 cans per hour, which of course you can see is about a can per minute, all day, every day, which is not physically possible, no matter how filthy and littered the city wherer you are collecting is. Sure you could
occasionally hit the mother lode where you find a large pile of cans in a corner or alley or whatever, but remember, our hero is
collecting litter, as you said, so he isn't raiding trash/recycle bins. He is walking the streets, from place to place, picking up a can at a time. No way you find a can a minute doing that. But assuming he does all that, at 78 years old, he would have a million dollars for his next of kin to inherit through probate, since he never had time to marry, have any children or write a will.
Again, this is a lovely thought exercise and I'd love to continue fleshing this out. I will look again at the rest of what you said and respond to that later, because you bring so many more interesting points in the rest of your post.