> companies with an oligopoly on most of the transfer of those notes
This is an important distinction. Credit card payments are not notes. They are not cash. So these companies have very little to do with the transfer of cash.
The point of "legal tender" is that if you try to pay off a debt in cash, they can't claim you haven't paid it and take you to court. If you try to pay for your meal in a restaurant with cash and they refuse, you can just walk out and they wouldn't have a legal case (probably. in theory. not legal advice).
What's tricky is this has to be a debt, as in past tense. If you try to buy groceries with cash and they refuse, you can walk out but you can't the groceries with you.
Participation in the cash market is mandatory on anyone who is owed money. Everything else on top (credit cards, checks) is essentially voluntary. Merchants can take it or leave it, the processors can come or go.
If you want to make an argument about the outsized effect that Visa has on the US monetary system, that's totally legitimate. It just has little to do with the concept of "legal tender".