Starting your own payment processor is done reasonably frequently, it has a certain barrier of entry (a few million and a year or two for all the approvals, so feasible with investor backing if you have a valid business reason to make a payment processor) but it's being done regularly and also you can buy and rebrand an existing one, which is a common way to get a quicker market entry at the cost of a bit more money.
However, that new processor will be a member of some existing payment systems e.g. Visa network and bound by all the same constraints as the other processors there. Creating your own alternative to a payment system (e.g. a major, widely and internationally supported card network like Visa or MC) is not really plausible. Perhaps you can look at the whole setup of cryptocurrencies+all the crypto exchange companies as something like such an alternative system, so there's one alternative made in this millenium.
However, that new processor will be a member of some existing payment systems e.g. Visa network and bound by all the same constraints as the other processors there. Creating your own alternative to a payment system (e.g. a major, widely and internationally supported card network like Visa or MC) is not really plausible. Perhaps you can look at the whole setup of cryptocurrencies+all the crypto exchange companies as something like such an alternative system, so there's one alternative made in this millenium.