While often the same bank performs, or is a partial owner involved in, both processing and issuing, the percentages you quote are largely disjoint.
> I'm reading that Visa typically charges 2.3% to merchants ...
The Discount Rate[0] a Merchant pays varies significantly based on the line-of-business, charge-backs, processing contract, and a bunch of other things.
> ... at least on average 1% of each transaction goes to cardholder bonuses ...
Again, issuer costs have no bearing on Merchant fees nor the profitability of processors.
0 - https://www.creditcards.com/credit-card-news/glossary/term-d...
If this is true, then where does the cash back come from if not as a cut of the merchant fees?
> If this is true, then where does the cash back come from if not as a cut of the merchant fees?
Interest rates + membership fees charged by issuers to Card Holders.
To use square as an example, card present is 2.6% whereas card not present is 3.5%.
Some of this is paid for by people carrying balances. The bet is that cash back will attract enough people to get and use the card that, in aggregate, interest will make up for cash back.