When you use your credit card or debit card in a store, you swipe it in a Point Of Sale terminal.
That terminal connects, via network, with payment processing software managed by Visa (and other card suppliers) to authorize the payment and move the money from your balance to store's balance.
If you were to create your own credit card, how would that processed? Visa is not going to process your credit card.
In order to have your credit card processed you would have to convince stores to upgrade POS terminals to also talk to your card authorization network.
That's just not gonna happen which is why Visa is not going to loose their lock on credit cards.
The only viable solution would be legislation that forces Visa to process other people's cards, maybe via some standard protocol, at a reasonable price (and that price would have to be much, much lower that Visa's fees to make that workable, because you want to make money on processing fees too).
I have a credit card from Wells Fargo. It's still made by Visa and Visa takes its cut of credit card processing fees.
My point is that in practice you can't issue a credit card and avoid Visa (or Mastercard) processing fees.
You would have to get most merchants to upgrade their POS terminals to support your payment network and that's not going to happen.
Technically there are 26 different cards most POS terminals support. 11 of those are Visa or MasterCard
Edit: "Private labeled" is considered anything a normal gateway will accept and go into the generic category. Which absolutely exists
Of the companies you mentioned, only Paypal could really do this, and in fact, they did have some in-store options available, although they're apparently being shut down. It's really a marketplace problem though -- you have to have enough consumers so that it's useful to businesses, and you have to have enough businesses that it's useful to consumers. And you have to have enough clout that the rest of the banking industry won't refuse to deal with you.