OK first off that checkbox isn’t smart. It doesn’t check any permissions or do anything special. When you check a box the program sends an AJAX request to the server to change a setting in the settings table of the wordpress db.
Now that checkbox is on an internal wp_admin screen so in theory it should just work. The fact that it isn’t for you tells me there must be some kind of plugin or something that is disabling the ajax request or not allowing your user to make a change to the settings table. Some security plugins stop those kinds of requests.
So my best suggestion is this: Temporarily disable all the other plugins (or at least a security plugin if you have one). Check the box in the settings and see if it stays checked. Then turn on the other plugins again. Nothing would change that setting once it is set so it may just work at that point.
