I have never had a problem using my Amex internationally either. I pay zero annual fee, get a cash back bonus. Internationally I do pay some kind of processing fee on top of the purchase price. There are cards (Amex and non-Amex) that do not have that fee. Once we start traveling again I will find one of those cards to use when I am out of the country.
I do carry a VISA for that rare instance where Amex is not accepted. I bet I use it less than once a month. I do have some recurring charges go on that card. That way if I need to shut the card down to stop those charges I am not nerfing my main card.
Of course I pay them both off in full every month, so I do not care what the interest rate allegedly is.
During our last trip out of the country we had a few places that would not accept AmEx. We used our other cards when that happened. The other issue, which is likely not that big of deal now, was the credit card should have a chip and pin, since this was used by many retailers in Europe. We had a few places that would not take the Visa or Mastercard because it was not chipped. I think all of our cards are chipped and have pins now.
One advantage of having good credit, and spending money on the cards, is service. I lost my AmEx card recently.

Went to a store, paid by AmEx, went to another store and noticed the card was missing.

Paid with a Visa and went back to the other store to see if they had found the card. No card so I called AmEx to cancel the card. They told me that I would not get a new card until the end of December which was not a big deal. Then Customer Service, said, "Oh, we can FedEx a new card to you tomorrow."

We put everything possible on the cards to earn airline points or cash back. Pay off the cards at the end of the month. The plane tickets for our next trip overseas will cost us very little because of the saved up air line miles.
Still not convinced the airline miles are better than just getting cash back though...