What I'm having trouble grasping is how more opportunities will give you better returns.
Again my experience is mainly with the stock market where diversification is good up to a point but after that the returns tend towards average.
Think by extremes. A guy with a $1000 account will have the same opportunities as Warren Buffet ?

If you still are not convinced by the drawing. Let's make an example with numbers.
Investor 1 has funds X and is short 1 on 1 given instrument. Investor 2 has funds 10X and is short 3 on that instrument. Investor 2 is allocating his (larger) capital to other instruments too obviously, some of which are also long protective legs.
You can imagine, as an example, we are dealing with put options, for instance. Now the market falls and the margin requirements for that option go up.
Investor 1 experiences a decline of say L due to the pure loss and, say, that the rise of margin requirements arrives at (X-L) so that he finishes the available funds. He is forced to close the position.
Available funds for Investor 2 are 10X - 3L - 3(X-L) + PNL variation of other instruments + Total Margin Requirements Variation. He has 7X plus the (PNL variation of other instruments) minus the possible increase of margin req on the other instruments.
Now, considering that investor 2 can have any other instrument and obviously he has probaly used some of his capital to buy long protective legs too, it turns out that the (PNL variation of other instruments) may even be a significant gain which adds to the 7X (or a small loss).
Also, the total Margin Requirements Variation may not be reducing significantly his available funds due to the hedging effect of long legs and the portfolio margin method.
So for all practical purposes, having more $$$, and a proper algorithm that uses it efficiently, obviously, will give you often the possibility to continue trade where "smaller fishes" have to give up or stop.
Real-world example. Investor 1 (top) was trading around 1MM (and he did not mention he had not selected the portfolio margin method).
Investor 2 (bottom) and a larger account and was using proper margining.
What happened? When the mkt went down, the first one was not able to ride the way up because many of his position had been either hedged (with unfavorable buy orders) or even liquidated.
So he delayed significantly his returns.
The algo will still carve inexorably a profit because it exploits a real edge, but the unfavorable orders have dampened it.
It's quite simple stuff. Nothing arcane
