Interchange. The fees go directly to the issuer not the network (which collects much smaller scheme fees). If you have 3 cards they’d almost certainly be for 3 different issuers so they’d split the interchange. Making you less valuable.

> for the rebates and profits

And salaries and pensions, etc.

In this context, it’s important to the merchants: They want these customers, so they grit their teeth on the higher interchange fees demanded by the banks. If the banks started handing out these cards to everyone, the merchants would revolt.