As far as damage goes, depends on load you placed on it when fluid loss occurred, distance travelled, and what temp the transmission/oil reached before you switched it off.
Just like motor oil, automatic transmission fluid (ATF) absorbs the heat produced by moving internal parts. ... However, if you add more strain to the gearbox, the components have to work harder, which creates more heat than the stock transmission cooler is designed to handle (which is what has happened when you lost some of your transmission fluid).
A transmission cooler helps the transmission stay cool, even when working harder than normal, which in turn lets you get the best performance and long life out of it, especially if you are towing
You can’t keep the auto fluid cool enough, and yes I know that a few vehicles were made without a cooler, many early American cars in 50/60’s had finned alloy transmission sumps to dissipate the heat, instead of a cooler.
If you don’t use cooler, it would have to be for a very short time till you either fix the cooler or replace it with and accessory cooler.
Check out this article below, should answer any questions you have.
https://www.transmissionrepaircostguide.com/low-transmission-fluid/Hi