It is not a case of the “NEC may frown on it”. Most local jurisdictions have adopted the NEC into law and technically if you violate the NEC you are violating the law. Now realistically what you propose will work under normal circumstances. I am surprised a reputable HVAC contractor would do that however.
The reason the NEC prohibits using the equipment grounding conductor as a neutral is that the return current causes a voltage drop so that the grounded parts of the equipment may be at some potential above ground. Practically it will only be a few volts but could be higher if a heavier load like a tool is plugged in to the receptacle. Using the ground as a neutral also means the path back to the neutral in the panel is no longer defined. Current can flow back through the ground conductor but it can also take other paths through the ground.
Oddly the NEC for years allowed one exception to this rule. Beginning during World War II it allowed the neutral of an electric range circuit to also be used to ground the range frame and did not require a separate equipment grounding conductor. An old time EE told me this exception was put in the NEC to save copper during the War. The exception was removed sometime in the 1980’s I believe and now you have to run a separate ground for ranges. Ranges used mostly 240 volts but many had 120 volt lights and a 120 volt convenience receptacle was common.
Edit: I checked and the exception was in the code until 1996 and included dryers too. Using a grounded conductor (neutral) as a grounding conductor. Range and Dryer Connection History- Wenatchee Home Inspections | NCW Home Inspections, LLC