You may be surprised to know that, when doing a "conventional" CC transaction, you are most certainly not giving any stranger information that would allow them to perform a transaction in your name on another merchant. What you are doing is providing your card information to a PSP (payment service provider) that has been contracted by the merchant and will provide the merchant with a token with which the merchant can trigger a charge request to your card but only to their own pre-approved acquirer account. The merchant can do nothing else with these tokens.
A breach of the merchant's token database would be embarrassing but harmless. A breach of the PSP's database of card numbers would be bad and inconvenient for the cardholders, sure, but it would be a business-terminating event for the PSP as its PCI DSS [0] compliance would be shattered and it would be unable to operate again.
In summary, ordinary card payments are essentially as secure as Apple Pay. The only difference is that in one case you are trusting a gigatech brand which is very saliently involved in the process but whose side-business in payments has only operated since 2014, while in the other case you are trusting businesses that you may or may not have ever heard of —Adyen? Braintree? WePay? Worldline?— but that have probably been dealing with secure payment processing as their primary or only business for much longer.
[0] https://en.wikipedia.org/wiki/Payment_Card_Industry_Data_Sec...