Bitcoin does NOT need to win the lottery or have some exorbitant pay off in order for it to be of considerable value. Actually, even if bitcoin maintains its value flat, it is doing a heck of a lot better than the dollar b/c the dollar is inevitably programmed to shrink in value due to dilution (too much expansion of the supply).
Sure, cash under the mattress is definitely a bad investment. On the other hand, the stock of a good company usually has a positive net expected value, typically around 5% of the price plus inflation.
So, you go wrong in a couple of respects regarding your bitcoin/lottery discussion, the first, as referred to above, is the anticipation that some major payoff is needed in order for bitcoin to be successful.
When I started paying attention to bitcoin, a few months ago, one of the two main "selling" points used to convince people to invest in it was the demand/supply argment, ostensibly proving that its price would rise to six figures or more when, at some not too distant future, everybody starte using bitcoins. (A few days ago someone posted a photo of a projected slide showing such numbers for various scenarios, ending with "investment/pension funds keep their capital as bitcoin" and some astronomical number beside that.) And in this thread, the "hodlers" apparently still believe in that math.
There were (there are) several things wrong with that argument, the main one being the implicit assumption that cryptocoin = bitcoin. A year ago, the salesman could jump from "cryptocoins are so nice that everybody will want to use them" to "bitcoins will be used in X% of all e-payments" without people noticing the gap. Back then, altcoins were only a theoretical possibility, which was dismissed by claims about the "Network Effect" and "First-Movers Advantage".
Well, since then altcoins became a reality, did not fail as predicted, and some of them seem to be strong competitors to bitcoin. One thing that bitcoiners missed is that any "Satoshi" who creates an altcoin can make a lot of money by selling his pre-mined coins at the right moment, even if the coin flops later on. Clearly the Network Effect and First Mover Advantage weren't strong enough to counter that "Copycat Advantage". Moreover, some altcoins do have some advantages over bitcoin, such as faster processing, democratic distribution of pre-mined coins, or a lethally cute puppy face.
Anyway, now bitcoin salesmen cannot do that jump any more. They must keep saying "cryptocoin" instead of "bitcoin" all the way to the final demand/supply formula. But since the supply of cryptocoins is infinite, not 21 million, the conclusion will be that "when cryptocoins are used to their full potential, the price of a cryptocoin will be at least zero." In other words, that demand/supply argument just went poof.
The sad fact is that now there is no argument or statistical analysis that will justify any claim about the value of a bitcoin in that ultimate future. It definitely may be zero, even if cryptocoins succeed beyond all dreams.
The other aspect that you go wrong is to expect that any investment in BTC has to be for a predetermined number of years. Markets have inevitably shown that investors are capable of playing their investments by ear and to go with the flow with whatever the investors perceive to be best to provide them with an acceptable return.
That is a very different "market", namely the speculative investors who aim to make money by buying and selling as the price changes. Indeed, because of its high volatility, bitcoin may appeal to those speculators who believe that they are smarter than their peers. However, short-range speculation is basically a zero-sum game. In practice, it is a negative-sum game, because of trading fees and time wasted. So, the expected value for a randomly chosen speculator is negative, like that of a lottery ticket.
And, anyway, that use is not what cryptocoins were invented for. One thing that the world does
not need is another financial instrument for speculators to play with.
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