That trope has a dual premise. First, compound interest is in play. Second, the beneficiary is not drawing on the account for living expenses.
If you can put $10 in an account at 3% interest and wait 200 years, you can take $3693.56 out. The key is not needing that money at all over a 200 year time span. The interest only accumulates on money that stays in the account.
Rich folks can afford to let money sit and gather interest. Poor folks spend all that they earn, if not more.
As a thought experiment, take two identical trust funds. Put $10M in one, and $20M in the other. Both grow at 4% per year. At the end of each year, the beneficiaries may withdraw up to 3% of the fund. The beneficiaries have similar tastes, and initially withdraw $250k. Each subsequent year, they draw 3% more, unless they hit the cap.
How many years before the larger fund grows from double to triple the size of the smaller? 25. It is four times the size after 42 years. It is 10x the size after 112 years. It takes 14 years for the less endowed beneficiary to hit the withdrawal cap, and 128 years for the other to reach it, at 10.64 times the size, spending 10.64 times as much per year, forevermore.
If your expenses grow more slowly than your investments, you will grow ever richer. If your spending grows faster than your income, such as for everyone in the US earning a wage or salary since 1970, you grow poorer.