We sometimes get asked if members can choose which merchant services provider processes their credit card, or if our “cashiers” can take manual payments.
Just like a pub, restaurant, or retail store, Giza utilizes a merchant services provider to be able to receive credit cards for recurring payments. Most merchants never indicate which provider they use, unless the provider makes the branding clear (e.g. Square tends to be obvious). If it’s FirstData at some store in the mall, you’ll likely never know. There’s also generally only one, because of the complexity and expense of maintaining two sets of charges to reconcile. When we buy movie tickets, we can pick what card we pay with (Visa, Amex, Mastercard) or (if it’s an in-person cashier vs. tickets online) then we could pay with cash, but in the case of using a card, only one merchant services provider is offered. The theater, pub, store does not have multiple credit card machine options, one tied to Wells, another to Chase, another to Square, etc.
Our membership system integrates with only four possible merchant services options, of which we likewise utilize only one. We had the option to set up Paypal (which we tried for a while, but which we haven’t found suitable to our needs, though there are some memberships from back in the day still grandfathered in. In fact, we have to routinely audit the system to account for the less-smooth integration in that first experience. We could have chosen ClickBank but it likewise did not seem right for us. The remaining two choices were Authorize.net and Stripe. Authorize.net is the granddaddy processor that’s been around for ages. You’re using it when you buy something from Purina.com or Parchment.com Stripe has been the least ‘corporate’ and most agile, we have a lot of experience with it, and it has, hands down, the best integration with our membership system. In five years, we’ve only had one significant hiccup lasting a week vs. many such hiccups with another payment processor, to the point we stopped signing up members that way.
So to answer the question ‘can members choose which bank or merchant services provider or payment processor processes their Visa or Mastercard’, no, that’s not an option.
CASH? We don’t take cash for the obvious reason, that we don’t employ cashiers or a physical storefront. But also, we’d not only need cashiers, but people manually tracking every membership and handling status based on cash flow. If there are recurring model businesses that work that way (Rotary Club, etc) they’re a dying breed and/or depend heavily on in-person interactions.
CHECKS? We don’t take checks for the same reason, and because they offer no additional privacy incentive. In fact the compliance requirements to which we’d have to adhere to handle financial information in a paper medium far exceed those of digital.
NUBUCOIN? We don’t take alternate or crypto-currencies, because the tools and resources we need to operate a community, not to mention utility companies and the like, don’t take those things. Starbucks does, and Amazon.com might, but we can’t shop there and pay the expenses of running the community in coffee beans and prime videos. 
Even were it otherwise, we’d lose a ton of people who don’t have the knowledge or inclination to own said coinage, so we gain nothing on popularity.
BARTER SYSTEMS: We can’t take chickens like a barber/dentist in the Old West. And we can’t require all members to sign up for some barter system. In the end, we can only take people who utilize the most common system in the world, US currency or a currency that converts to it, in a location-independent format (i.e. digital).
As long as we pay our taxes, adhere to the business rules and conventions of the United States, etc, we’re working in dollars, and the dollars are digitally processed and digitally backed. And, in the end, all payment processors operating on US soil (including cryptocurrency firms trading here, regardless of where they’re located) are required to share certain information with the US Treasury and are routinely blocked if they don’t.
For our purposes, therefore, there is no such thing as anonymous money. If financial transactions with US firms looks that way, it’s an illusion. We’ve identified no sustainable way to operate without using the US financial system, and no means by which that system provides for anonymous transactions without violating US law, despite appearances and, at the present time, we’re not willing to operate in breach of the law.