Is the only way to use this card though the Wallet App on my iPhone?

No. You can use the physical titanium Apple Card anywhere that Mastercard is accepted and in the same way that you would use any credit card.

See information in How to use Apple Card - Apple Support including

You can pay with your titanium Apple Card anywhere in the world that Mastercard is accepted.

and also see the "Pay with your card number in Wallet" section of that link for information about paying online where Apple Pay is not accepted.