If yours does, I would still read the fine print and compare with ones on insuremytrip. I picked a random CC and read its benefits, and did not see one involving serious illness - only death, accidents, and things like flight delays.
Edit: Checked again: Apparently mine does provide travel insurance. But as expected, not for serious illness (e.g. stroke that requires medical evacuation - happened to someone I know during travel). Also, it only covers issues related to the mode of travel. So if you're in the Bahamas for 2 weeks, and you get into accident a week in, that is excluded. With insuremytrip plans, the whole duration of stay is covered.
See: https://www.americanexpress.com/us/credit-cards/features-ben...
Too lazy to look up whether Amex provides for problems during the trip (beyond evacuation), as opposed to merely during travel.
or get it free through your credit card. Many premium cards offer it as a free benefit.