RC,
Nope, you're confusing the layout, which is understandable since it's a little convoluted.
Browning is owned by FNH in Herstal, Belgium, as you noted.
"Winchester" has been owned by FNH for several years also, as you noted.
The FN-USA plant in South Carolina is owned by FNH in Belgium.
"Winchester" as a fully fleshed business entity has not existed for quite a while. The USRAC plant in New Haven was not Winchester, by the time it was closed it was only a plant owned by FNH that produced three models carrying the Winchester brand. Those Winchester-branded products were sold through Browning channels, and serviced & repaired by the joint Browning/Winchester center in Missouri.
Browning has handled the Winchester product line behind the scenes for some time, with both "companies" headquarters located in the same building in Utah.
FNH had Browning buy the Winchester licensing rights so that FNH (through FN-USA in South Carolina) can manufacture "Winchesters" in the US, and can manufacture Winchester marked guns outside the US & import them as "Winchesters" through Browning as the import agent.
FNH, through its three subsidiaries in the US can produce "Winchesters" in this country in South Carolina, at the Belgian plant, and/or have them produced by Miroku in Japan and other plants (Turkey, Russia, etc.) .
Browning has been actually doing the marketing of Winchester guns from whatever source they came from, and the companies are all inter-related to some degree in that sense.
It's something of a shell game, but since Browning actually manufactures nothing itself & was already importing many of its firearms from other countries and handling the "Winchester" brand anyway, positioning it to buy the rights to the brand when they came up for bid does make sense from FNH's point of view.
Denis
Edited to add: Not sure you'll find many Winchester shotguns made in South Carolina, I have not looked for a few months but I think you'll see most are produced in other countries.