At first, I was under the impression that this proposal was specifically about the disambiguating suffixes in parentheses, but I think I still agree with it even if it’s about moving the full constructed name out of name, as long as it’s done carefully.
To elaborate, one reason for naming relations differently than the member ways is that only the member ways represent the physical infrastructure and contribute to an address. The roadway can have a cardinal direction in its name that represents the quadrant in a street grid, whereas the route has a different direction indicating the direction of travel. We can still name a freeway name=North Interstate Highway 35 even as it carries southbound Interstate Highway 35:
Many of the relation names date back to when a shield-capable renderer or route-aware router was still aspirational, so nothing was actually using network. But since then, routers such as OSRM and Valhalla and renderers such as OsmAnd and OSM Americana have added sufficient support for network that the constructed names are really only for the benefit of editors. Yet it actually gets in the way of editors potentially synthesizing a fuller label for the relation, because the name could be significant for all it knows.
Nominatim’s developer has cited constructed and descriptive relation names as the reason why it doesn’t index route relations: