I think what you’re saying should work. To test, I’ve done the following.
In my parent theme Twenty Thirteen, I created a folder called TEMPLATES and put a standard page template in there (new-page.php) called “NEW”.
Created a test page that would use this particular “NEW” template.
Then created a child theme called (twentythirteen-child), that also has a folder called TEMPLATES, that also contains new-page.php. Same as the original new-page.php, but with a green background.
Then I activated the child theme, and made sure that the test page would still use “NEW” as its template.
Turns out it DID take the new-page.php from the child theme! So this leads me to believe that as long as the folder name, file name and template name is the same, it will take the child theme’s version.
Maybe you’d need to republish the page?