Well, there is theory, and there is real world.
In the real world, reductions in debit card fees (in the USA, via the Durbin Amendment) have not lead to lowered prices, just increased profits. [1]
This explains why merchants care about processing fees.
[1] From page 27 of
https://www.law.gmu.edu/assets/files/publications/working_pa...
"In the year following Durbin’s implementation, gasoline retailers realized more than $1 billion in savings from reduced interchange fees. But 'while this should mean savings of roughly 3 cents per gallon, no savings have been passed on to consumers.' This is particularly remarkable and instructive given that the retail gasoline industry is highly competitive, and at least one other study has demonstrated that gasoline excise taxes are almost instantly and fully passed through to consumers"