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I'm not sure what you're saying. Is it that "route" is derived from "router" rather than the other way around?

No, just that "route" the verb was derived from "route" the noun relatively recently (OED has no reference before 1893 for the verb, whereas the noun is C13th). Once "route" became a verb, there was nowhere for its gerund to go except "router", a different pronunciation of which was already in use as the gerund of the much older verb "rout", also C13th.

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Certainly in US English, "route" has existed as both a verb and a noun since before the Internet, usually referring to roads and directions.

Before the Internet, yes. Before, say, the telegraph -- no.

Peter