I can't speak to how 2025.1 is, we have a late release date and I haven't really gotten into it yet. However, I believe I can deconflict what the help is saying: My interpretation here is that running a search unrestricted should show all fields/results even ones a user doesn't have access to, but the permissions on the record type of a search are not overridden by this setting. An example: You have a custom field on the vendor record, that is restricted to a particular role. Now you run a vendor saved search and include that custom field and mark the Run Unrestricted checkbox. A user with a role without at least the Vendors (view) permission would NOT be able to run the search at all and thus would not see the search results. However, a user, who does have this permission, even if their role is not able to view that custom field, WILL see that field in the results in the saved search.
As a corollary, I believe the expected behavior is that joins should generally show up with the Run Unrestricted checkbox as long as the user has permission to the primary record type for the saved search itself. I.e. permissions control whether or not you can actually run the search, but run unrestricted will show all the results assuming you can run the search. But who knows what support will say?