I don't think people actually agree on the definition of capitalism itself, I just looked it up and was a little surprised how definitive it is:

an economic and political system in which a country's trade and industry are controlled by private owners for profit.

If you asked whether capitalism is a political system, at least in my random polling, 2 out of 9 respondents said No.

True, because it's also a giant Ponzi scheme. We pick up new debt today to pay off debt from yesterday, and we hope expanding GDP and inflation will always offset the difference.

The difference between a pyramid scheme and a good business is where your money comes from, in a pyramid scheme it comes from the people at the bottom of the pyramid, in a business it comes from selling goods and/or services, that's not saying I agree with big business, but one is profiting off of legitimate customers, the other is profiting off it's own "employees". I nearly got caught into one a few years ago too, until I realized what it was, at that point they had only taken a couple $100 for the interview and sign up stage, i had to block my card for them to never get access again, because even though i didnt complete sign up, thwy kept charging me monthly

Now it’s the owners of the holding companies who own the owners of all the rest doing the controlling.

Except that you're describing commerce, not capitalism. Capitalism is the idea that the commerce has value, and that one can own the value of the commercial activity. Further, the value of the business is tied to it's growth.

Further, products and services are never exchanged in an equitable transaction between two parties, because capitalism necessitates a third party, the capitalist. The capitalist must acquire products and services from employees for less than their true value, and then sell them to consumers for more than their true value.

And because capitalism demands growth, one or both of those two margins must continue to expand. This means workers must be pressured to work for less and less, which is why the capitalist opposes social services, universal healthcare, and affordable housing. This also means the capitalism opposes consumer protections, environmental protections, and taxes that provide a functioning society that might interfere with their growth.

Now what happens when every producer and consumer is fighting for the same margins, the same advantages, and the same growth? Then the capitalist seeks new avenues for new capital and new capitalists. Building business on ensuring the growth of business for other capitalists. Selling the idea that you, too, can join us at the top of the pyramid, all the while kicking down ladders they climbed to get there.

So no, the system itself isn't a pyramid scheme. It's just an idea that encourages pyramid schemes because it relies on impossible growth, like a cancer eating away at society.

Transactions are often not equitable, and most wealth bubbles up to the people on the top tiers.

an eventual lack of ability to recruit more people

Global population growth is slowing and predicted to halt within the next few decades.

The modern economy has always relied on an environment of growth, which will now end. In some ways this resembles a pyramid scheme; in the past anyone making investments in the economy could count on a larger new generation of people coming into the market needing to exchange their labor for those hoarded resources. Since this is no longer going to be the case, new entrants to the market are going to be getting a lot less for their effort than they may have expected based on past trends.

Please explain, how exactly is fractional-reserve banking a pyramid scheme?

If we take into consideration the destruction of the ecosystems necessary to sustain human life, capitalism is a net-negative.