This is a good start, but I would add a few of details to fill out the picture.

1. It is important to distinguish more carefully between “income” and “wealth.” Wealth means assets minus liabilities. It is more unequal than income, not just because the wealthy have more assets, but because the less wealthy have more debts in relation to their assets. You can find some charts and links here: https://medium.com/swlh/left-behind-working-class-trends-in-wealth-are-even-gloomier-than-trends-in-wages-f5ccf41c782d
2. Also, take your example where the rich guy and poor guy both earn 5% interest, but the middle-class guy has less capital to start with. True enough, but reality is worse, since they don’t earn the same rate. The rich guy can probably get a 10% return in some hedge fund where the middle-class guy will be lucky if he can get 2% on a CD from his bank, and more likely gets 0.01% on his checking account (if he even has one of those).
3. Yes, I agree it is very important to provide more opportunities for poor people to improve their economic position. But it is also important to eliminate the unfair ways that the rich have to get richer, not through their own productivity, but by gaming the system (rent seeking). For example, if you are well enough off to own a home, the first thing you do is vote for politicians that support restrictive zoning, so that no one can build new affordable housing. The value of your own home goes up and those not yet on the housing ladder are priced out of the market. Only one of many examples. More here: The Captured Economy How the Powerful Enrich Themselves, Slow Growth, and Increase Inequality (Lindsey and Teles) https://www.amazon.com/Captured-Economy-Powerful-Themselves-Inequality/dp/019062776X