There are two possibilities - create a channel in the trim or create a channel in the floor.
For network and TV cables, you probably need at least a gap about 1/2 inch wide and 1/4 to 3/8 deep. The trim looks too shallow to put in that channel. You could cut a channel like that and cover it with the existing or comparable trim.
While a router can cut channels, it needs several inches around its cutting bits for the base of the unit, and your cut would be too close to the raised hearth.
A good alternative might be a multitool
These are specialized power saws that can cut in very close quarters. The saw blade (it has several others) is offset, so you can cut downward into the floor right up to the edge of the hearth. They come in various price ranges starting at about $60.
I would remove the molding, draw a line about 5/8 out from the hearth all around. Then I would cut downward through the floating floor using the half circle wood blade following along the line. You probably will need to use the straight saw blade at the corners.
The wood strip you cut off should be easlily removed if it is not glued down. If it is, a small prybar, cheap screwdriver or chisel should easily pop it out.
Lay your cable, replace the trim. You may want to use cable clips to keep the wire in the channel until the trim is in place.
I think those gaps in the floor are part of the charm of old wood flooring. I definitely would not slosh epoxy all over them. I guess it partly depends on the style of the house and the type of buyer you're trying to attract. Original wood flooring may be attractive to some people if it's restored nicely.
Best Answer
Looks like either there's no subfloor under that board, or the subfloor failed (rotted?) in that area. If this is ground floor, it might be worth looking at this area up from the basement or crawl space or whatever's under there, to see what was supposed to be supporting that board and what condition it's in.
That's assuming that there was something there. Someone may have done a sloppy install over a hole and simply hoped that nobody would ever put weight on it.
If you can get to it from underneath, and it was never properly supported, I'd be inclined to see if I could push it back up into position, then nail some support in place under it -- one or two 2x4's between the joists, tight up against the floor, then insert shims to fill any remaining gap between those and this board. Not very elegant, but more elegant than leaving a board floating in midair.
If you can get to it and discover that it was originally supported but Something Has Gone Wrong, you'll have to proceed from there.