In general, route relations aren’t critical to routing but can be useful to navigation systems on the margins, as can highway classifications. They can help mitigate the unsolved routing problem of unintentional divebombing, hint at simpler routes and more intuitive guidance instructions, and of course present visuals that match the signs on the ground. Those visuals are just about the only thing that navigation systems want to use the network information for. Apart from that, some routers like cycle.travel use network=* as a heuristic in context to mitigate some other undertagging, but these heuristics are often based on coincidences or probabilities, rather than the first principles that might guide tagging discussions.
Nominatim doesn’t even index route relations in the first place because mappers have treated name=* as a junk drawer and the PTv2 scheme tried to make modern art out of that junk. We’re only beginning to climb out of this mess.
That’s a long list of things for a single network=* key to mean. Of these things, the symbols are the most in need of a single key that can unambiguously direct the renderer, the only thing that’s too objective for heuristics yet critical for wayfinding. Displaying a symbol that isn’t misleading requires either semantic network information or presentational details like a full machine-readable description of a logo. If you thought the Grand Rounds Scenic Byway has a complicated logo, you ain’t seen nothing yet!