What additional US-power or ICAO requirements apply to a departing US-based crew with a US-registered aircraft flying to land in Mexico?
The only aviation-related requirements on crossing the US border are:


All flights crossing the US's Air Defense Identification Zone (ADIZ) in either direction must be on an IFR or DVFR flight plan, talking to ATC and have a discrete transponder code.
A list of all passengers and crew must be filed with eAPIS at least 24 hours before crossing, in either direction, and the plane must have an eAPIS decal.


The only difference between leaving vs entering that I'm aware of is that you don't need to leave from a designated port of entry. (Some other countries do require that.)

As a practical matter, there are a handful of aviation-related things that apply to most/all other countries that US pilots don't have to worry about in the US:


The pilot(s) must have ICAO-recognized license(s). FAA Student, Sport and Recreational licenses are not.
The pilot(s) must have appropriate medical certificate(s). No BasicMed.
The aircraft must have a normal airworthiness certificate. No experimental aircraft.
The aircraft and at least one pilot must have FCC radio licenses.


Technically, you don't need passports to leave, but you'll need them to go anywhere else (and to return) by air, so they are effectively required for both.

There are thousands of other laws that apply to all people or goods leaving the US regardless of mode, such as not exporting restricted technology or weapons, human trafficking, fleeing prosecution, etc.  (And there are even more for entering.)  I doubt anyone could list them all, and that is probably off-topic anyway.