I think the concern is somewhat misplaced, because we've already seen the FCC take multiple stabs at regulations on net neutrality, that have been based on the principle of no or limited interference with lawful content, and they haven't required inspection of traffic.

Its not a requirement that ISPs block unlawful content, its a prohibition on blocking lawful content. A strong enough prohibition on blocking lawful content (e.g., one which provides consequences whenever lawful content is blocked, even if the intent was to block unlawful content and the lawful content just got caught up in the net) actually reduces the incentive for ISPs to attempt to identify and block unlawful content, since attempting to do so but making a misidentification in the process can result in penalties.

In fact, I'd argue that a no-blocking-lawful-content rule combined with a safe harbor provisions preventing ISPs from being liable for transmission of unlawful content by customers so long as the ISP wasn't actively involved in soliciting or promoting the specific illegal content is probably a good protection against ISP-initiated snooping, since it eliminates any liability-based incentive for it.

OTOH, it doesn't stop ISP-snooping-at-behest-of-government, but if you get a mechanism to do that, its going to be outside the context of net neutrality.