I can’t imagine that tight patches could ever approach the friction caused by a .223 bullet.
Keep in mind, though, that a really tight patch can cause your cleaning rod to bow and rub against the bore with some force (assuming you’re pushing the rod, not pulling it).
I wouldn’t see this as much of an issue with a one piece, coated rod. I think it might be more of a problem with a segmented steel (or, heaven forbid, aluminum) rod.
Note that patches vary quite a bit from one manufacturer to another. I can’t imagine that you’d ever be able to get a thick mil-spec .308 patch down a .223 barrel.
The exact size and type of your jag/loop can also affect the fit of the patch.
Also keep in mind that, while the bore may appear clean and shiny, you really do need to occasionally use a bronze bristle brush to clean carbon out of the edges of the rifling.