What I was saying is that at the time I didn't know.
I was assuming that if the box stays ticked that that means that the change does get submitted, even though returning to the site settings page fails. That doesn't necessarily have to be true, as It's possible that the status of the checked box (checked or not checked) is not pulled from the same place in the database(s) that the status of the OOS on or off variable that eBay uses to manage the GTC listings, but unlikely since that would be a stupid way to handle things.
I don't have any GTC listings, and thought I couldn't test anything, but after thinking on it a bit, looking at the developer API documentation, I ran some tests, and turns out that OOS also applies to non-GTC variation listings.
https://developer.ebay.com/DevZone/guides/ebayfeatures/Development/Listings-UseOutOfStock.htmlThe setting is working for me even though the return to the settings page fails after submitting the change in the same way you described. So yes, my guess was correct, and toggling the setting in site preferences does turn OOS on and OOS even though the "page not responding" error occurs when the return to the settings page fails after submission.
Note: My simple tests* indicate that that site setting OOS change will not apply to current listings unless the listings are revised to get the change to "take".
*added a variation to listing and saved while setting showed OOS enabled. Everything normal. Revised and changed qty of that variation to zero, saved, and it showed OOS. Changed site setting to not use OOS, and looked at listing, which still showed OOS. Revised listing and changed price by a few pennies, saved, and the variation disappeared as it should when OOS is not enabled.
If you want to change a bunch of GTC or variation listings to use OOS, they will need to be revised to get eBay to go grab the new OOS setting and apply it. You could bulk edit the listings after changing the OOS setting, and just make a change to something in them (bump or reduce price by a penny, find/replace something in the description - like add a space or period to some unique word in all the descriptions, etc).
You can test your listings to make sure the OOS changes is applying by revising one and reducing qty to zero. Should show OOS on the listing itself, and disappear from search, or if a variation listing will show OOS on the variation.