OK - so alt text isn't sticking when using the action. Maybe try to add some manually, saving, simply to see if it's the action that's not behaving well, or the file itself.
Call up your Accessibility Tools, and choose the Reading order tool. Find a figure in your doc, and click on it's small reading order number in the upper left corner. Now that it's selected, right click and choose Edit Alternate Text. Add the alt text for just this one image. Save, inspect, and close the file... then re-open. If the alt text 'stuck' I'd then try using the Set Alternate Text tool in the Accessibility tools to assign alt text to the rest of the images. Inspect - save - close - reopen & inspect.

I would be interested to know how this turns out. Also, is this issue with just this file, or do others behave the same way?
My best,
Dave