Interchange fees primarily fund consumer rewards programs and benefits. To become an appealing choice for consumers, any new payment method has to offer competitive benefits. Those benefits are funded by the fees.
Visa/MC/AmEx have essentially created a system whereby higher-spending customers are able to wrench more value from merchants in the form of higher fees. This is reflected in the fee schedules that slice and dice merchants by category and customers by card tier.
If you want to build a new payment system it is important to understand that it’s not just a negotiation between merchants and issuers. It’s a two-sided market where customers also leverage their spending power, directly or indirectly.