Sinclair pays the TEAMS for the broadcast rights, not the leagues. That is how it works in the three leagues in question. Local rights belong to each team. That is why they say "Copyright, Cincinnati Reds Inc." (or whatever) at the end of local broadcasts and "Copyright Office of the Commissioner of Baseball" at the end of national ones.
So Sinclair, et al, produces a game for the LOCAL broadcast on an RSN. So now these get "nationalized" in two ways. One is the packages, like mlb.tv, NHL Center Ice, or MLB Extra Innings, and the other is that some get shown on MLB Network or NHL Network as I type I am watching Bally Sports Michigan production of the Twins @ Tigers on MLB Network.
Sinclair, et al, is paid a VERY SMALL fee for the use of these local productions nationally, it has never itself shown out of market games itself.
I like SBJ, and its covers a subject I have great interest in, but it really could use some editing, fact checking and proof reading. I remember a very long article about baseball's international presence in Central America. Lots of quotes about a team in Panama City.
Unfortunately they were about the one in Florida.