Yes, Shift management reports only record cash payments, good enough for the cash accounting at the end of the day.
But in the back office you can see a full report about card payments as well, in the "Sales by payment type" report. This should help you to manage credit card sales.
And you can actually create your own payment types, you could add a "Gift Card" payment method, and track gift cards as a tender. Again, you could follow the reports in the "Sales by payment type" section in the back office.