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Not specifically, its just something that isn't used much. In actual fact I don't know of an application of Tor in my experience.
Routers have the capability of doing what you say the servers would do, so can you not set them up for that type of routing of traffic?

Perhaps my explanation was not so good, I will have to go back and read. The Tor nodes are computers all over the world. The information is bounced around through several different nodes with different IPs to mask the originating IP.

As for application. The technology was originally created by the United States Navy as a way for its overseas operators to relay information back without being tracked.

Journalists use the Tor network quite often for anonymity. Intelligent whistleblowers like Edward Snowden have used Tor for anonymity.

The most beneficial uses are not by just average persons surfing the internet. Although Tor is the only way to access the .onion sites, otherwise know as the infamous “Dark Web”.