More specifically: if you're in the business of processing payments on behalf of others, you actually have "bank" as a core competency / requirement of your business model and need to make your plans accordingly (or carry the added risk of having your core business model outsourced).

What magic wand do banks possess that enables them to declare-into-existence a merchant account, anyway? I mean in a technical sense, not a legal sense. What are payment gateways and other banks seeing, that allows them to know that a particular merchant account X is a "real" account, that can be targeted with credit card payments, rather than a made-up one?

Is there an X.509-based hierarchical bank registrar system for charge-origination signing certificates, for putting charges onto the card networks? Is there a DNS registry of merchant accounts for pre-checking charges before attempting them? Do banks underwrite other banks into existence by signing their certs?

Pretty much none of that. What exists is a whole lot of money, a willingness to lose that money if a bank makes bad bets on the trust and security of a customer, a bunch of laws to adhere to, and willingness to go to jail if those laws aren't followed (or, perhaps willingness being the wrong word, an understanding one will go to jail if those laws aren't followed and protected parties lose money as a result ;) ).

In short, it's a different category of risk than the category that an individual business or even a business that is acting as a middleman for transactions takes on. And it's a different category precisely because most of the solutions aren't technical; they're legal, social, and financial. If a bank gets screwed it just gets screwed; even if the law intervenes to deal with transgression, money is often just gone as a result of such fraud. So they are comparatively more conservative in their decision-making.

Banks have a whole lot of bank laws. While mostly it is paperwork, it also means some trust and responsibility that can be used to do things that banks do fornon banks like process money. Check with the local laws. The law protects customers of banks at the expense of the bank, which is what you want customers to know before using you for bank like things.

Yes, this is why my previous company never went the IPSP route (letting customers accept payments with your merchant account). They are incredibly arduous to get approved, you practically need to have a bank CEO as your godfather to get it. Also you need to be at least PCI Level 1, which involves actual auditors going through your business and policies. That part is significantly easier than the IPSP though. OP doesn't sound like they were trying to do that though. They talk about their client's individual Stripe accounts being turned off.

This is probably what a business like OP's would need to do. When their customers are small, use a processor like Paypal or Stripe. But as customers get larger, OP should probably do what we did: partner with an ISO, who can get the customer their own merchant account. OP still does the processing for them, but the risk and finances run directly through the client, not OP. The ISO can also add in a margin on the transaction fees for OP if that's part of their business model.