A la Glenski: An employer would invest in his/her employee's development of skill in order for that employee to provide greater value for his/her company.
In an ideal situation, yes. In Japan, that just doesn't happen. Men have a hard enough time getting paternity leave, let alone getting paid for studying to improve themselves. The vast majority of Japanese companies don't chip in a single yen to help their J employees learn English when it's needed, either. So, why would they have a mind set that says they will help a foreigner (whom most companies don't even consider permanent)?
I know of several people in the US who have had their M. Ed. paid for by a school or the public school system in exchange for teaching labor at the school. I would be seeking an similar sort of arrangement in Japan if at all possible.
Good luck, but I don't think you'll find any.
Now, in terms of long-term marketability and job sustainability, which degree would hold more value? I'm considering the markets for both the US and international education. A Masters in English Education from a place like NYU (to teach in Secondary Schools and above) or a Master's degree in TESOL?
Can't help you with any teaching career advice outside of Japan. If you want to see what Japan wants, go to the JRECIN web site and see what sort of qualifications employers ask for in terms of degrees. Keep the following in mind:
1) Employers here don't even know what a CELTA is, let alone a degree in TESOL.
2) English is not a second language here. It's a foreign language. ESL vs. EFL. This might affect the outlook on TESOL.
3) Most teaching jobs at unis are for part-timers. This is related to the degree requirements as well as any benefits that you wrote about.
4) To move up the academic ladder in unis, you need to have very good J language skills (including reading/writing/speaking) beyond mere conversation because you would be required to serve on committees and do various administrative work.
5) Everyone that I know who has done a distance degree while working in Japan has continued working here and has not gotten a single yen from their employer to help out.
If I go the TESOL route, I'm thinking about doing something like this: 1) study for the TESOL while teaching Eikaiwa at night or teaching business English; 2) work for a company like WestGate while studying for the TESOL. I would be gaining more experience teaching adults in either situation. Would working for a company like WestGate help me get my foot in the door? How would the extra experience and qualification help my chances working for a uni?
Universities would laugh at anyone who has only eikaiwa work under their belt, and that includes Westgate. Doing Westgate work is just teaching eikaiwa-like classes on a campus, and if memory serves, those classes aren't even accredited. Some teachers put a real spin on describing that kind of work, though, claiming it's actual professorial uni work. What a joke.
Would you please let me know what other sort of employment opportunities are available to me in Japan with an MA in TESOL?
Just teaching English. There are more opportunities than eikaiwa and university, of course. ALT, private HS, kosen, business English, your own school.
Vincent and johnnyG have given you sound advice, too.