I would vote no on this idea, in part for some reasons you have already raised.
To my eyes this would be not just unnecessary site clutter (scenic byways are mostly redundant with existing systems anyway - mostly), but also headache inducing for any traveler attempting to finish clinching a jurisdiction, who would now be expected to grab some random segments here and there of county and local roads that happen to have been designated part of a byway.
Also worth noting is that byways tend to be signed only supplementary to the numbered routes they run along, and signage is often infrequent or spotty. There is for example absolutely no signage at all for the Lake Ontario Circle Tour, and the Lake Erie Circle Tour is not signed in New York or Ontario.
I am of the mind that we should be sticking, at least within North America, to numbered roads on the state highway level or greater, with only occasional exceptions for other roads that are major enough to warrant inclusion (e.g. NY Parkways, the occasional freeway out there with no signed number)
That said... if we end up implementing a site feature where users are able to "mute" systems they aren't interested in tracking travels on, then I would withdraw my objection.