How are they trying to export the data?
If I follow your example, after clicking the layer to access the layer group, it takes me to the item details for that hosted feature layer. Once there, there is an Export Data dropdown that gives several options. However, I don't see any that actually take me to the REST point as you second pic.
I only get that if I try to get to it with the URL at bottom right of the detail page.
If you are trying to get access to that (using URL), you can't just copy/paste the url anymore as it will substitute your org for the services1 server. If this is what you are after, you need to click the "View" link next to URL,
then, when the services open, right click on the link to server and open in new tab/window. This will get you to the service with credentials and won't get the token error.
R_