For a long time, before browning meat that would then braise, I've coated it in flour as to ensure the right consistency for the resulting sauce. Are the downsides to this method? Should I just brown the meat for braised dishes without the flour coat, then thicken the simmering liquid with a slurry?
One potential downside to this method is that with a thick coat of flour, you're mostly browning the flour, not the meat, and thus possibly creating different flavor compounds than if you were searing the meat directly. Maillard reactions are complicated stuff. If you're doing this, you should probably shake off excess flour to leave a very thin layer so that you still get browning on the meat itself.
The related problem is that you're less able to accurately measure the amount of flour, less able to make sure it's easily browned, and thus less able to control the overall balance of thickening power and flavor it contributes. Roux (flour cooked in oil as a base for sauces) gains additional flavor, but loses some of its thickening ability as it cooks. You could monitor this over the surface of your meat, but that goes against the typical wisdom of not moving meat around the pan as it browns - that is, you can't see how browned the floury surface is getting when it's face-down.
Personally, I'd prefer tighter control over both of these processes over the convenience gained by browning the meat and flour together. You can easily brown the meat, remove it from the pan, then add the flour and some additional oil to create a separate roux without losing much time. A slurry will work too, but takes plenty of cooking before it loses its raw, grainy flavor.
I have done it both ways successfully. Flouring the meat before browning does give a little extra flavor, plus the flour continues to cook while the meat is braising. I personally think it gives a richer, deeper flavor and ensures that you won't have any raw flour flavor. If you are happy with your results using this method I wouldn't change it. :)