Will other payment method no longer work if their icon is not shown in the product page?

My online store also offers Cash on Delivery as a payment method. If I use the app, does this mean I can no longer offer COD to my customers and they should only choose one of the icons shown?

No, of course not. Your store can still offer and accommodate other payment methods with no problems at all. The trust icons only indicate the guaranteed payments made on your online store and do not necessarily mean that other payment methods will no longer work or be accepted.