Leaving aside other inaccuracies in the model, and leaving aside insulting non-responses like "that's a lot of words" that inherently penalize analyses of a complex system and favor oversimplified models even when incorrect...
The cherry-picking in question, here, is that you are choosing a model that claims the rewards being paid to richer people are a subsidy of the rich by the poor, by attributing one input to one output, while ignoring everything else in the system. Think about double-entry accounting for a moment: would it be reasonable to point to one inflow and one outflow and say "this inflow paid for that outflow!" as a serious model, without looking at anything else in the system? And in particular, would it be reasonable to claim that as the "core point" and demand that people refute that point while ignoring any evidence invalidating the oversimplified "one inflow, one outflow" model that privileges that point and treats it as the core point?
To draw an analogous argument, which is faulty for the same reason the credit card rewards "subsidy" argument is faulty: if a store offers a "bulk discount", is that a subsidy from poor people (who may not be able to afford to buy or store in bulk) to rich people (who on average can do so more easily)? Using the same modeling you're describing would paint it as such: poor people pay more for the same goods. I would argue that it's more accurately modeled as an incentive offered by the store that they see as on net bringing in more revenue? Do you expect that the store, upon receiving that higher revenue, needs to use it to somehow lower the prices paid by poorer people to compensate? And if they do not use it as such, and instead pocket that additional revenue or use it in some other way, does that suddenly make this a subsidy from poor customers to rich customers?
I am, in general, in favor of the argument of analyzing a system's net effect and not just its intentions. But it's faulty to analyze a subset of the system and then attribute fault or foment outrage based on that simplified model of the system.
If you want to argue that the entire system is not the perfect system, or that it could be improved, you'll get no argument from me. If you want to argue that, in general, "having money is a massive advantage for getting more money", again, that's entirely true right now.
If you want to argue that you've shown a specific subsidy of the rich by the poor, I think you haven't given evidence to support the validity of that model. And in particular (and to my mind the more important aspect of that argument insofar as the point of arguing about systems is to evaluate potential changes to them), if you want to argue that in a different system with no interchange, poor people would pay less for the same goods than they do in this system, I think you'd need more of a model of the overall system to successfully argue that. In particular, among other things, 1) would merchants actually make more in that alternate system rather than making less (which the observation that merchants make more money from higher-income customers calls into question), and 2) would merchants actually pass that savings on, specifically, to lower-income customers, which the observation that they don't currently charge different prices for different customers calls into question, and 3) would some other part of that system change as a result that makes things worse in other ways, such as products like cards or checking accounts (a checking account is a credit instrument) no longer being available to people with lower income or lower credit scores?
I'm not making this argument because I think the financial system is perfect and shouldn't be changed, or that the financial system overall is fair. I'm making this argument because the oversimplified model that foments outrage by claiming a subsidy of the rich by the poor is a bad model for the purposes of reasoning about the system and attempting to make it better. Criticizing that model and arguments based on that model isn't based on wanting to preserve or advocate for the status quo, it's based on wanting to accurately model the world as a step towards evaluating potential improvements.