"reach the river" - It's okay, but there's something about it that seems off-center for some reason. I think using "reach the river" makes it sound like the river is a far-off destination, or the river is on the way to a far-off destination. So, yes, it's okay, but for some reason, to me, it sounds like it would be reserved for certain contexts, whereas "get to the river" and "come to the river" sound general or "more regular".
That's my take on it.
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