would it interfere with the signal in some way?
It will only interfere with the signal if you kink or squash the cable. If you put the cable straight through and do not press down on it, it won't be a problem. Note that it can be difficult to avoid pressing down on cable underneath something on which you walk (e.g. the threshold). I do not know if there is room underneath that threshold where the cable would fit. You might have to pull up the threshold to see.
Coaxial cable is a regular, usually solid, wire inside a hollow conductor (often called a hollow wire, it looks more like aluminum foil). The inner wire and the outer wire are separated by a dielectric (a fancy word for an electrical insulator). The two wires share the same central axis, thus coaxial.
My point is that so long as you don't kink or otherwise squash the wire, the outer conductor will continue to protect the signal on the inner wire. Squashing it can cause the two to touch or just make the dielectric too thin to be effective. This is why the things that they use to secure coaxial cable do not squash. It's also why you don't use regular staples, which do squash the cable. Avoid having the threshold push down on the wire.
The current version (that you find unsightly) is safer, because you don't walk on or over the cable. You also might consider running the cable higher. Perhaps if it were at the ceiling, you would find it less unsightly. I personally would find that less unsightly than running it in front of the "skirting board" (baseboard is a more familiar term to me).
If you have access to a basement, crawlspace, or attic, you could run through there. However, the couple times I saw the current method used, it was in a condominium or apartment building, so I'm not terribly hopeful about those options.
What are these called
When I search for any of
- raceway
- cable channel
- cable management
combined with plastic or vinyl, I find those. Unfortunately, I am the wrong person to ask about removing the trim or installing the raceway. My useful experience regarding this question was from installing cable modems. For more complicated installs like you describe, I would have referred them to either an electrician or a specialist.