Thickening with flour
Unless you are making a vegetarian braisé, I would suggest not browning the vegetables.

If you brown the vegetables and add them too early in the pot, then they will cook too much and be mushy in the end product.

I prefer not using flour when browning the meat too early because I tend to burn it; so I brown the meat, and after that, do a roux and re-add the meat and add liquid.

As for using beurre manié, for me it is just a thickening agent and is done at the end and should not really add additional flavour to the dish.