I don't understand how the stache (say a pure index fund stache) will actually grow at all if we set the fund to send us a check for dividens, capital gain/interest.
You are conflating two different things. The Trinity Study is based on withdrawing 4% of your portfolio, adjusted for inflation annually. Since the stock market, on average, tends to do better than 4% + inflation, over time, your portfolio will grow in most scenarios.
What you are focusing on is
how you withdraw that 4%. One way to do that is to have your dividends and annual CGs sent directly to you -- but that number will very likely not end up being 4%. In most cases, it will be less, because usually dividends are on the order of 1-3%, and realized CGs are much less than that (you just get those when the fund you own buys/sells shares, so it tends to be minimal). So in most cases, to get to your full 4%, you will need to supplement by selling some of your investments.
You are correct that, if all of the companies you invested in passed along 100% of their profit every year as dividends and/or CGs, your investments would not grow. But per the above, most companies/funds do not do that -- they reinvest some of the profits, which increases the value of the business over time. And even if they did, that would also very likely mean you were withdrawing much more than the 4% used in the Trinity Study -- as mentioned above, the stock market tends to do much more than 4% on average, and that is because most companies turn more than 4% profit every year. So if a company decided to pass on all of its annual profits to shareholders as dividends, and you had them send that money directly to you, you'd be pulling out way more than 4% in most years.
It helps keep things straight if you keep a mental line between the
amount of money you can take out every year (the 4%/Trinity Study) and
how you take out that money (which is where dividends and CGs come into play).