I'm an Italian citizen and got my ESTA last year that's still valid until my passport expiration in 2023. Italy is also in the 6 months club so it shouldn't be a problem.
I will travel to the United States, in May but my passport will expire in August. My only concern is: Is the 6-month cutoff a general rule or could the border agent let me in anyway?
Your question as written doesn't make a lot of sense.
As you've stated, Italy is on the list of countries that do NOT require an additional 6 months passport validity when entering the US. This list, which is commonly known as the '6 months club' is documented in the US Department of State regulations as '9 FAM 403.9-3(B)(2)' which means that it's an official rule, and thus not something generally open to interpretation.
Thus as an Italian citizen you will be allowed entry to the US even if your passport does not have 6 months validity beyond the expected end of your stay. Note that if your passport has less than 90 days validity then your entry to the US will also likely expire at the same time as your passport (ie, you may not get the full 90 days that someone entering on ESTA would normally receive).
If your request was more generic, such as for a country that is NOT in the '6 months club', then the answer is likely "it depends". Immigration staff do have some ability to override requirements like this under specific circumstances, however it extremely unlikely that you'd even get the chance to present your case - checking passport and visa requirements is generally carried out by airline staff (and/or automatically by various systems on check-in), so if you didn't meet such a requirement then it's highly likely you would be denied boarding and not allowed to actually travel to the US.
