This is basically what Capital in the 21st Century said too. People who reinvest money they make into making more money are getting richer. Those who spend it are not. Pretty straight forward.
Except that unless you are lucky, you're unlikely to become wealthy from just saving/investing ordinary income. You're both presenting it as if it were a simple choice between being frugal and engaging in consumption, but it's not.
There's more than that implicit in my comment. If you're the sort of person who views wealth as a scorecard rather than a means to consumption, why should you care whether you end up becoming wealthy or not? You'll be too busy making money to spend it. And if you're the sort of person who views wealth as a means to consumption, then of course you'd like to be wealthier, but, well, you know how to achieve that.
(In actuality, the distinction isn't binary - most people desire both consumption and accumulation of wealth, in different proportions. But that reinforces the meta-point I'm trying to make, that money is a means to make choices about your life, and what makes those choices meaningful is the fact that there are constraints in the first place.)
Not to be argumentative (because I think we have somewhat similar opinions on the broader topic), but I am having real problems with your scorecard analogy. I have certainly met people who think that way but a) of course they want to be wealthy, because being in the game but having a low score isn't satisfying and doesn't come with any status benefits and b) even people who are obsessed with money-as-score still like to consume, partly to signify their ability to do so. Furthermore, almost all of the people with the scorecard mentality that I've encountered come from financially privileged backgrounds. I remember one person in particular who was working as a professional investor that I occasionally did some IT work for, and who confided in me that he sometimes felt it was an uphill struggle because he wasn't his father's favorite son and so had to start his investing career with only $1 million of capital. I forebore from mentioning that that was about 30 times my annual income and I would be delighted to swap my problems for his, but it was pretty dispiriting; while I don't view capitalism as a zero-sum game, there are far fewer investment opportunities open to people who have only tiny amounts of capital.
I agree that money can be a means to make choices about one's life, but only past a certain point. Poor people need money to obtain the necessities of survival, and that often doesn't leave a whole lot of options open for making life-scale economic choices. Picture a Monopoly game where some players get the standard $1500 at the outset and $200 each time they pass Go, others get $150 and $20, and one person gets $15,000 and $2000. No matter how good the players in the second group are, they're probably going to perform poorly under those conditions; likewise whoever is fortunate enough to start out controlling large sums is considerably more likely to win.
We all have the same amount of time at our disposal, and how we use that can certainly have a huge impact on our economic futures. But large capital disparities arguably provide a disincentive to maximize productivity insofar as people feel hard work will have little impact on their prospects for advancement relative to their contemporaries.
Hardly anyone makes more than $2M in ten years. Investing $1M conservatively and moving to the third world might get you a reasonable standard of living, but that's not what people mean when they say rich.
This seems to contain a number of assumptions about income, non-discretionary living expenses, and rates of return. You'll certainly be richer but you're not likely to become rich (which I guess I should have defined earlier, but which I consider to be wealthy enough that you can retire and live off passive income should you so choose).