Three-pronged, grounded type outlets should not be substituted for ungrounded outlets unless (1) a ground wire is connected to the outlet, or (2) a Ground Fault Circuit Interrupter (GFCI) Breaker or outlet protects the outlet.
(Figure 3, outlet option) Installing a new ground wire may require a licensed electrician to install a new wire to the circuit breaker box and may be prohibitively expensive.
I do see his reasoning....just seems like it is not an upgrade...it were me, I'd run new wire up through the attic..is more work, but if it is a ranch style house and you have an assessable attic, it would be do-able.
It's your call.....probably dependent on how much you can do, and how much you would pay someone else to do it.
The image on the left depicts a GFCI breaker: the distinctive indicator is the “Test” button mounted on the breaker.
An inspector may want to “trip” the GFCI in order to identify that the power shuts off to any ungrounded outlet that is protected by the breaker.
To “trip” the GFCI, the inspector will press the test button and the switch will move and shut off power to the circuit.