I have a few questions about cycle routes and OSM:
If some routes or sections are missing in OSM but are available from official sites, is it legal to upload them to OSM, either as-is or after (uselessly) redrawing them myself using online tools such as Cycle.travel?
I don’t know the situation in France, but, anywhere in the world, OSM can only use data that has been obtained from information on the ground (e.g. signposts), seen in person, or is derived from sources that have given an explicit licence that is compatible with using the data in OSM. It is possible that the people who define the routes have not given such a licence, and, as a result, coverage depends on whether or not a local enthusiast has mapped them based on what they can actually see along the route.
is probably a bug in the tool you used to extract the route. You have neither identified that tool, nor the route, sufficiently for me to check it. Although, ideally, route relations are sorted properly that, is often not the case, and even if it is, people may make edits that break the sorting, although JOSM will try to avoid that for changes like splitting and combining ways.
What I think has happened is that the tool you are using to view the route has inserted false segments to join up discontinuities. If you where to enter the relation number in openstreetmap.org, I think you would find that the route looked more sensible, with none of the long straight line segments that I think are artefacts of the tool that you are using.
In theory, for any route that was both complete and had no branches, other than two way branches that re-merge, for one way sections, it would be possible to create a tool that sorted the route relation mechanically. JOSM has indicators that help you do this manually, and I have done so for several bus routes, but I’m not, currently, aware of any automated tool for sorting route relations.
You should look at the relation to confirm that this is the result of its not being sorted, then raise a bug report against the export tool pointing out that is generating phantom ways for unsorted routes.
Creating a GPX file relates to the tool that you haven’t identified. It is not the aim of OSM to provide end users with GPX files; it provides the XML files of the relation for use by third party tools that can render the map as a bitmap, generate GPX files, etc., and some sample output as an aid to mappers.
You can add missing sections by editing the relation. JOSM has the best support for relations. You can copy from other OSM relations, providing that you have followed the route and are therefore certain that the routes overlap. Otherwise, you need to take account of the considerations in the first part of my reply. In general, you should only use a third party source if its is officially listed as acceptable on the OSM Wiki, or is included as an overlay in one of the standard editors.
Even if the licence appears correct, you should get official confirmation from the French OSM community before importing any machine readable definition of a route from a third party.