The leverage is that the activists will potentially be able to draw the ire of the government. Visa and MC get away with absolute murder in terms of the size of the fees that they charge in the US. Most developed countries don't allow that. The US government could easily regulate them (as they already do with debit card fees) or use anti-trust law against the obvious duopoly charging exorbitant prices. Because of this situation, Visa and MC have a very strong incentive to crack down on things the government doesn't like.
The unspoken arrangement is that the government allows them to keep charging a de facto sales tax on a massive portion of the economy as long as they cooperate and de facto ban things that the government wants banned but can't ban themselves due to that pesky constitution.
The unspoken arrangement is that the government allows them to keep charging a de facto sales tax on a massive portion of the economy as long as they cooperate and de facto ban things that the government wants banned but can't ban themselves due to that pesky constitution.