Lets begin here
I'm a US Citizen, working for a US company which pays 100% of my
health insurance costs with a US company. I'm living in Mexico, which
has "Social Security", which offers access to the government
hospitals.
I am assuming you are referring to IMSS which is related to the universal coverage aspects in Mexico vs Seguro Popular which is primarily for the uninsured and can be bought without being part of the "Social Security" system in Mexico. I do provide some basic information in that regards towards the end of the answer though.
Consulting this article which provides broad information about healthcare in Mexico and various options associated with it. It suggests that IMSS which is the Mexican universal coverage is not just due to "living" in Mexico. One of many ways you can qualify for it is working for a Mexican company which pays taxes and thus paying in to the system. Or there is an option for foreign residents to buy it.
Mexico’s Social Security System
Mexico’s social security system is called the Instituto Mexicano de
Seguro Social, often abbreviated as just IMSS.
Mexico’s Social Security System is free at the point of delivery for
Mexicans as well as foreign nationals with residency status who are in
full-time employment by a company registered in the IMSS system
(payroll taxes cover healthcare – see below). Foreigners resident in
Mexico who are not working (e.g. retirees) or not working for company
enrolled in IMSS may elect to purchase the IMSS health insurance
separately for a modest monthly fee.
On to the next part of your question:
Aside from avoiding the hassle of sending medical bills to my US-based
insurance provider (which covers "normal and customary" expenses, even
outside of the US), is there any advantage to becoming registered with
Mexican Social Security?
In the case of an emergency, such as a car accident, would I receive
better or different care if I was a member of the Mexican Social
Security system?
No direct answer there but it depends. Its advantages include the vast coverage it has. Although the quality of care varies due to a lot of factors:
Speed and Quality of Care
There are waiting periods for non-emergency procedures, and IMSS
members who get their coverage as part of their formal employment are
given priority over those who enrolled independently.
It can also depend on where you are in Mexico and what facilities are around you:
The reported quality of care varies, and the experience you have will
likely depend on where you are in Mexico and what the wider local
demand is on health services when you’re seeking treatment. Some
foreign residents report good care from IMSS, others report
disappointments and shortcomings. It’s fair to say—as with all large,
publicly-funded healthcare systems world-wide—that the demand for
services usually exceeds the supply of resources available and
compromises must be made.
Note that IMSS could exclude certain pre-existing diseases and there can be a waiting period in which certain conditions may not be covered. Refer to its website for more details or here to learn more.
Preexisting Conditions
Some preexisting conditions are not covered and these include
malignant tumors, congenital diseases, chronic degenerative diseases,
addictions, mental illness, and HIV—among others. If you have any
preexisting ‘excluded’ conditions, you cannot enroll into the IMSS
insurance program.
Other specific preexisting conditions are covered on ‘deferment’ and
these don’t preclude you for joining the program, but are subject to
specific waiting periods before you can seek healthcare services
related to them.
Also note:
Your IMSS insurance does not cover eye care, dental, elective
surgeries (e.g. plastic surgery, weight loss), infertility treatments,
or treatments for self-inflicted injuries.
SEGURO POPULAR
The reading suggests that this may be an option you may want to consider. There is a special FAQ here for foreign residents in Mexico considering this.
Seguro Popular is an option for foreigners residing in Mexico, it is a
medical service offered by the Mexican government for those
individuals that do not have with any other form of state issued
medical care such as: IMSS, ISSSTE, SEDENA, etc.
The following I would suggest is subjective, although it does seems you will need a good handle of Spanish based on the various reading I had.
As you can see, Seguro Popular is another good option for health care
in Mexico. Sadly most of the centers have mainly Spanish speaking
staff, so be prepared to use your Spanish or ask a friend to go with
you and translate.
There is also a nice blog written from a while back though but seems to echo the various points that may helpful to review before making a decision