I know of a few looths using this method - although I'm not sure what size U-channel you're imagining.
They use something about 5/8" or more wide. From what I can tell, they skin the insert, which is usually spruce, then embed that into the channel. Dion James and his shop-mate Tuna-Tone do this.
Yes - Dion does this, as does Burton LeGeyt (who I think introduced the technique to Dion). TunaTone learned it from Dion for sure.
Mike Kennedy / Indian Hill Guitars also does this. We all know each other - it's a really small world.
My approach is different as detailed above, since the insert in my case is not a straight piece but rather a fully carved conical shape, the same basic geometry as a neck.
The insert is 1/2" thick at the center - with a nominal 1/4" fretboard, that leaves .075-.100" of exterior "shell" after everything is carved to shape. When carving necks with exterior shells made from some woods (mahogany, e.g.) I sometimes discover epoxy in the pores that has soaked through from doing the CF layup.
The idea is to put the CF as close to the back surface of the neck as possible - this is where it has the greatest mechanical advantage.
It's great advantage is its tensile strength, balanced by the excellent compressive strength of the fingerboard.
With those two aspects in play, you can make the middle of the neck out of something nice and pretty-sounding - redwood, spruce, basswood...just about anything, honestly.
Even Balsa would probably work, or foam. The Klos carbon necks are foam-core.
Cheers,
Martin