He's totally wrong
Your house is connected correctly. The inspector is completely wrong and he's been told that before, and he chose to huff out his chest and say "well I'm the inspector and that makes me right".... rather than have either the concern for work, or the sense of wonderment, to spend two minutes to learn the technology he is inspecting.
So yeah. Fire that guy. What he's really saying is "I want to see the buttons so I don't need to go out to the car to get my GFCI tester and then ask you where the reset is". This lunacy is also why the big box store sells GFCI receptacles in 10-packs.
...And serving regular receptacles downline from a GFCI device is correct. Not least, it protects the wiring behind the receptacle, which matters if you splash dirty water onto it.
In order to be legal, your downstream receptacles must have a "GFCI Protected" label. He would be correct to write you up for their absence.
Okay, there's one place where "faster" (as in lower threshold to trip) might make a scrap of sense, but only in Europe. There, they are fond of 30ma GFCI devices, often for whole-house protection (of the structure from ground fault related fires, not the people from shock). 30ma is too high a threshold for personnel protection - it can kill the weak, and knock anyone out cold, causing death by secondary effects such as drowning or falling. So yeah, if you're in Europe, and your downline GFCI device is 30ma not the proper 8ma, then okay. In the US, we use AFCI for fire protection, and most GFCI devices are about personnel protection.