If an outlet has a ground terminal, it should be grounded. No ifs or buts. (And IMO, giving the illusion of ground protection is even worse than not having any protection.). I don't think you can even get two-prong outlets any more, so I think you'll have to fix the wiring. It sounds like you're still negotiating, so this is a good bargaining point. And if you don't get it fixed now, when it comes time to sell, any potential buyers would be within their rights to make you fix it then.
An easy way to check that your wiring has a ground conductor is to open up the service panel, look for the breaker that controls those outlets, then trace the black-shielded wire to where the cable feeds into the panel. (If it's present, ground is unshielded.)
If you have grounded wiring at the service panel, the next thing to check would be to look inside the outlets in question to see if the wiring there is grounded (it may just be a loose connection, and since it's common to daisy-chain outlets in a room, one faulty wire could affect several outlets).
If you have don't have ground at the outlets, but did at the service panel, you probably have a junction box joining the older ungrounded wiring and newer grounded wiring. Time to go exploring in the attic or crawl-space / basement.
GFCIs detect a difference between current going out on the live terminal and current coming in on the neutral terminal. If you're the easiest path to ground in an electrical fault, a GFCI will detect that and break the circuit; without a GFCI, you'd have to hope that you draw enough current -- and for long enough -- to trip a regular circuit breaker. On the whole, though, I'd rather that current was running to ground through some copper that's intended for just that purpose.
National Electrical Code (NEC), provides the answer in article 406.4 (C).
406.4 Receptacle Mounting.
(C) Receptacles Mounted on Covers. Receptacles mounted to and supported by a cover shall be held rigidly against the cover by more
than one screw or shall be a device assembly or box cover listed and
identified for securing by a single screw.
In your example image, the receptacle mounted using only one screw is a code violation.
Best Answer
If I am seeing your photos right, it's a mild security feature. When I worked in electrical contracting, that was done so that a panel could have a lock installed on the door, and then (theoretically) prevent unauthorized access by surreptitiously removing the panel front.
Like most locks, they are meant to deter rather than completely prevent access. If someone wants to get in, they will, but the unauthorized access will take some effort and will be detected.