It is a general truism but the reasons are that the rules change dramatically when you simply have more capital.
Here are some examples, limited to particular kinds of markets:
Under $2,000 in capital
Nobody is going to offer you a margin account, and if you do get one it isn't with the best broker on commissions and other capabilities. So this means cash only trading, enjoy your 3 business day settlement periods.
This means no shorting, confining a trader to only buy and hold strategies, making them more dependent on luck than a more capable trader.
This means it is more expensive to buy stock, since you have to put down 100% of the cash to hold a share, whereas someone with more money puts down less capital to hold the exact same number of shares.
This means no covered options strategies or spreads, again limiting the market directions where a trader could earn
Under $25,000 in capital
In the stock market, the pattern day trader rule applies to retail margin accounts with a balance under $25,000 and this severally limits the kinds of trades you are able to take because of the limit in the number of trades you can take in a given time period. Forget managing a multi-leg option position when the market isn't moving your direction.
Under $125,000 in capital
Worse margin rules. You excluded portfolio margin from your post, but it is a key part of the answer
Over $1,000,000 in capital
Participate in private placements, regulation D offerings reserved for accredited investors. These days, as buy and hold investments, these generally have more growth potential than publicly traded offerings.
Over $5,000,000 in capital
You can easily get the compliance and risk manager to turn the other way on margin rules. This is not conjecture, leverage up to infinity, try not to bankrupt yourself and the trading firm.