Why wouldn't the Tigers go after starting pitching this off season?

They probably should, but it isn't their top need. They need to pursue an outfielder who can stabilize one of the corners for them, and get some major bullpen help. If they have money for a free agent SP they might get one, or trade for one. The Braves have a SP surplus and need an OF I could see a Boesch for Beachy scenario working out.