Very hard in 2018 to get a card without any Visa, Mastercard, Amex, logo on it. Thinking about it I am not even sure if it exists.
What is not allowed is to use the logo of Visa Mastercard on a website if you have no agreement/relation with the companies or a 3rd party (which is using Visa or Mastercard)
I heard that debit cards with function of payment in crypto currency already exist and almost every new crypto project has in plan to produce their own debit cards for payment. Here is the latest one which I found:
https://bitcointalk.org/index.php?topic=3251590.0 Well, I am not a profy in crypto world yet and for now I can't imagine how such monsters of the market as Visa and MC will let to survive to such project, but as I love to say the time will show.