Someone with a lot of wealth owning stock doesn't impose a constant expense on anyone with little wealth. Who owns what stock has no effect on how much it costs me to live; what businesses exist and competition between them and so on does, but someone is going to own each of them at the end of the day and opening that ownership up to a broader variety of people doesn't make any difference (other than the overall positive economic benefits of investing money with those best capable to use it and diversified risk and all of those things that stock enables).
Stock is also an investment that's immediately available to anyone who has any kind of savings income to set aside, unlike something like renting income properties which requires you to already have either huge savings or a huge income just to get into that game (to buy the extra housing) and then gets easier from there. So stock exhibits none of the iniquities that I'm concerned about. It's a form of investment that everyone can participate in.
In fact people with lower incomes benefit the most (proportional to their invested amount) from stock investment, if we assume that their lower incomes are indicative of an inability to generate a return on capital through their own activity. By instead trading that capital to someone better capable of generating a return on it, in exchange for a portion of their business, everyone including the lower-income investor benefits more than if they had just hung onto that capital and not invested it. Whereas someone more capable than average of generating a great return on capital through their own actions, who would likely have a higher income already because of that productivity, is better off hanging onto that capital and putting it to good use, rather than investing it into other people's less-productive ventures.
Also, I never said anyone shouldn't be able to borrow money. Interest-free loans are possible, and if useful will still exist in a world without interest. I imagine a roundabout form of such (payment plans) would be extremely common in an interest-free world: expensive items can be bought on different terms (payment due dates) for different prices, where you're charged a higher price for longer terms (more delayed payment). A pair of such transactions is how temporary use of expensive things could still be possible without technically renting them: the "rental" company would technically be a buyer and seller of used [whatever]s, selling on longer terms for higher prices and buying on shorter (immediate) terms for lower prices, and the "renter" would technically be buying something on long terms at a high price and then selling it back on short (immediate) terms at a lower price well before he's finished paying it off, thus canceling the remainder of his debt toward the purchase.
He'd lose some money in such a the deal and the "rental" company would profit from it, but the "renter" could technically avoid that by selling back to someone else for a better deal... if he was willing to accept the inconvenience of finding such a buyer, which most people wanting to use something temporarily wouldn't be, so the profit to be made "renting" things out in such a way would be driven down the the value of the convenience such a "rental" company provides. Meanwhile people who don't want something temporarily, who want something permanently and just can't afford the huge barriers to entry into ownership, can start "renting" and just keep it up until eventually it's theirs and they can stop, and if they do eventually decide they don't want it (they're moving house, for example), take the time to find a buyer who will take it off their hands for a price and terms that doesn't leave them losing on the deal.
The tricks that Muslim (and Middle Ages Catholic) countries use(d) to get around the prohibition on interest usually involve(d) a rental contract as part of the complex financial instrument used, so those would not be technically possible under my proposal. That was one of the problems with such earlier opposition to interest: it didn't recognize it as just one (major) instance of the broader phenomenon of rent which is the source of the trouble. (Another problem is that they actually banned it as in you would be punished if you tried to enter into such an arrangement, whereas I only advocate not legally protecting such contracts, which makes them unsuitable for widespread economic use [as what good is an unenforceable contract], by having the state do less rather than more, maintaining compatibility with libertarian principles [in the broad, not-US-centric sense]).