The brand on your card is heavily influenced by what your bank is offering; your bank (card issuer) is offering products that benefit themselves; and they benefit from higher interchange, not lower interchange.

If Visa or Mastercard would say '0% interchange' then that would make them not competitive - banks wouldn't market those cards much and wouldn't offer any rewards/points on them; merchants would benefit but customers wouldn't, since merchants wouldn't be allowed to give discounts to the 'cheap card' anyway - the rule is 'same price or you can't take our cards at all and you'll get less customers'.

Technically, they aren't. If you're willing to start a new credit card company, and get enough people on board, you could compete on interchange fees. However, how are you going to convince the consumer that they want a credit card issued by you?

Large associations of merchants (who'd benefit from lower interchanges) can do that, IIRC there are some examples of successful country-local cards. Making any of it an international network comparable to Visa/MC would anyway take decades and billions.