The $10 off $50 is merchandise, not restaurants, so I’m assuming you’re asking about a merchandise discount?
According to the Disney Visa discounts page, you have to use your card to get the discount. If you want to do self checkout, then you need it linked to your
MDE account. Otherwise it doesn’t matter if it’s linked or not. Using a rewards card with your points loaded would get the discount too.
The small print from that page:
Merchandise Savings at Disney Theme Parks & Resorts
Offer and offer elements including, but not limited to, participating locations and eligible items, are subject to availability, capacity limitations, closures, and additional restrictions and exclusions, may change or be canceled without notice, and are not guaranteed. A $50 minimum pre-tax single transaction merchandise purchase is required to receive discount. Must use your valid Disney Visa Card and/or Disney Rewards Redemption Card as form(s) of payment at the time of purchase for full amount of purchase and mention this offer.