Well that's the reason why the toukaidou line was considered the right one, it's because it means "the way to the eastern sea" and "to the sea" is the last thought Eva had before realizing what the river is.
There is another thing that sounds similar and that's Hokkaido: the way to the northern sea. However hokkaido is a whole region not a railway ^^;
What I don't find right in your reasoning is thinking that the village and the hometown are the same thing. Imho it is more like this:
Hometown = starting place
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Village (of gold?)
That's because the epitaph starts with the hometown and then if you follow the river downstream "yagate" (finally) you'll find a village.
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