What is the primary purpose of a table read, for say, a weekly sitcom? Is it so the writers/producers can see what works and what doesn’t? Is it so the talent can walk-through the script together? Or for some other reason? And does the production meeting usually happen before or after the table read?
All of the above. You often do the same thing for a play (though the author is not involved): in the first rehearsal actors read all their lines at a table so everyone gets a feel for the character. For a sitcom, the actors can get an idea of the lines and for possible business with the other actors, and, in TV, the writers/producers can more easily judge what doesn’t work and try to improve it.
Not sure about when the production meeting is held, but I suspect it would be both before and after.