No, if you have other accounts that don't qualify* for the Cash-back deals program, you can still participate. The card you choose to use needs to be activated and from an eligible account.
Since the program is based on your eligible card usage and participation, it doesn't matter which one you use. Just know the reward is credited to the account the purchase is made with.
*There are two factors that would disqualify it. Either the personal card being used isn't activated, or it's a business account.
- Business debit and credit cards aren't eligible to participate in Cash-back deals.
- The personal debit or credit card needs to be activated before you can earn cash-back deals, and they won't populate if the card is inactive.