This is about networks, not owners or operators. A network is a set of routes that operate together as a system, whether through interconnectivity or consistent branding. As a rule of thumb, a given route number doesn’t repeat on multiple distinct routes within a network, though there are exceptions. A given agency may be responsible for managing multiple networks (so we tag the agency as the operator=* of each network’s routes), and various other agencies may maintain the physical infrastructure that carries these routes (which we tag as the operator=* of individual cycleways).
I focused on the standard green ovals to avoid overwhelming this discussion. The vast majority of bike route shields have nonstandard designs. This makes it all the more important for some renderers to attempt to simulate these designs. The good news is that most of these designs, once simplified, probably will not be subject to copyright protection. Ironically, it’s only those federal Reuleaux triangles that we need to be careful about until we get ahold of the right officials.
Tentatively, I agree with you that these separate networks belong in cycle_network=*. It’s an unfortunate key name, but it’s what we’ve got for now.
If only.
The osmc:symbol=* scheme has many problems. It symbol selection is explicitly eurocentric (“hiking paths used primarily in Germany”). Very few of the symbols ever occur in North America. It doesn’t even provide for a bicycle pictogram, by far the most common element of a bike route shield, let alone the ingredients necessary for describing the hundreds of local network pictographs that occur on those oval signs, let alone the probably thousands of nonstandard shield designs across the U.S. Adequately expressing this diversity of designs would be a massive distraction for mappers who just want to map.
Until very recently, the wiki insisted that osmc:symbol=* was a proprietary scheme, take it or leave it. Though the wiki no longer defers to the scheme’s original author to such a degree, it still isn’t clear whether we can ever extend it beyond the current set of shapes. As a workaround, some mappers have tried to turn the wiki itself into a rendering engine using wiki:symbol=*. This idea is a nonstarter in practice.
You’re absolutely right, the renderers should be responsible for drawing their own graphics. Fortunately, this is what they’re good at! Renderers do not need mappers to encode alt text for route markers. They need us to interpret the markers for them. By analogy, when you see a 
sign, a router wants you to say that bikes are prohibited, not that it’s decorated with slashy-bikey ornaments.
I realize you don’t care about road routes, but I ask you to consider the innovations that are taking place in that department. Many renderers already draw their own shield graphics for road routes. Some are limited to a simple set of generic shapes, while others like OSM Americana aim for higher-fidelity designs. Each implementation has a different artistic style. Beyond shields, routers and geocoders use network=* to construct a systematic name for a numbered route. The format of a systematic name usually has nothing to do with the shield’s appearance. It may vary by language and diction.
This diversity is good for users. It’s only possible because road mappers set network=* to a presentation-independent value. Based on this value, the renderer chooses a template image from its own repertoire and fills in the ref=*, recolors the image based on colour=*, or adds a pictograph based on name=*. The OSM Americana project publishes a freely reusable repertoire of over 180 template images plus a dozen basic shapes. That’s just for highway routes, and we’re far from finished with those. I expect hundreds more once we’re through with bike route shields. The contents of these SVG files don’t belong inside the OSM database.
Never is a strong word. Many biking and hiking route shield designs incorporate text in case the symbols are too obscure. I think this is the Continental Divide Trail, just a guess:
And I’m almost positive that this is the Mount Carbon Loop Trail:
But you’re right, I have no idea what this sign is for – it must be the Lollipop Route. 