Agree with all the above. Will just add, consider the following:
There's never any 100% iron-clad guarantee that anything you ask for in a rental car will be delivered. Including what kind of transmission. Car rentals are just a squishy business with some things that just get sorted out (or don't) on the fly. That is, you can reserve an automatic, show up, and they don't have an automatic for you. The chance of this happening is lower at major airports and big city rental offices, but could happen at any of them.
Because of #1 above, you might want to have a "Plan B" in your back pocket for what you would do if you get unlucky with the transmission lottery when you show up to pick up your car.
Consider that billions of people around the world have figured out how to drive a car with a manual transmission. It's not very difficult. You could always do a little training and learn to drive a stick at home, before your trip. This would free you from limitations in the future. There are places in the world that you might want to visit someday where an automatic transmission car will simply not be available.
Driving a stick is (or can be) actually fun, and offers a better driving experience in some cases. It's not something to be intimidated or limited by (but does require you to learn how to do before trying it for the first time in a foreign country, where driving is already something of a challenge).
Good luck.