The authority, exactly
No job offer and no sponsor are required. Eligibility instead turns on where you graduated: your qualification must come from a university on the Home Office's own eligible-university list, which it bases on global university rankings and updates itself — UK universities are not on it. You may hold this visa only once, and the guidance states plainly that time spent on it does not count towards settlement. The same post-Brexit rule applies: an EU/EEA citizen without EU Settlement Scheme status needs this route (or another one) like any other non-British, non-Irish national.