Why on earth would you be philosophically opposed to a degree? Its a tool that you can use to get the type of job in Japan that you want, if you can get one then go for it, otherwise your options are pretty limited.
The main problem for you is that the normal avenue for non-degree holders, the working holiday visa, isn't available to Americans. I say go finish that 1/3 of the degree you've got left.
Also, about that "4 year degree" requirement, if you've got a 3 year degree then they'll still take you. I had a 3 year undergraduate degree from a Canadian university and I was hired by both GEOS and AEON despite that fact without any trouble.
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