Interchange was originally set up to encourage card issuers. This was because there was a lot more money to be made by banks offering payment acceptance to merchants than there were issuing cards to customers. You can't charge customers for having and using a debit card, but you can charge a merchant for accepting one - so the interchange attempted to address that imbalance.
As you rightly said, it's also been used to fund card loyalty schemes and Apple Pay also takes a small cut of that if they are involved in the purchase.
The biggest issue I know of relating to interchange fees is those of Amex which are typically larger than the rest of the market and used to fund their more generous rewards scheme. As a result, a lot of merchants (here in the UK at least) refuse to accept Amex.