The phone agent was correct.
I'll try and explain as succinctly as I can because these things can get tricky

- Medical cover: You get this regardless of how you've paid for your trip (amex, mastercard, cash, magic beans, buttons,...)
- Other insurance coverage: Things like travel inconvenience, cancellation, abandonment etc need to be paid using any amex card that you are the account holder (eg your BA amex)**.


**You can claim on your policy for purchases/travel not paid on your amex but only if the vendor does not accept or otherwise refuses your card and not because you didn't want to use your amex. For example, Loganair doesn't take amex but I'd still be covered in the event of a claim (only difference being it requires a bit more admin on my side to claim and best practice is that any subsequent purchases related to the claim are on amex). However, if you had an attractive points/cashback offer for BA on a Visa card which you used instead of amex, you would not be covered because the vendor accepts amex.