What is the advantage of a roux over a raw flour slurry, in sauce?
Roux Method

The advantages of the roux method:


It can be prepared in advance
The raw flour taste is cooked out when the roux is prepared, so the sauce is ready as soon as it is thickened; this also makes it easier to add more roux to adjust the thickness of the sauce.
It actually requires less supervision.  You are actually being overly fussy with your roux based sauce.  You could add all of the milk at once, although starting with one smaller batch just to dissolve the roux is a good idea.
The butter coats the flour particles, making lumping quite unlikely
Can be browned for additional flavor at the cost of thickening power


It also adds oil or butter to the recipe, which may or may not be an advantage.

Slurry Method

The advantages of the slurry method (which is what the second method is, although it is more typically done with water or stock than milk are):


It is fast and convenient, if you don't have roux prepared ahead
No oil or butter is required, so it doesn't have to be accounted for in the recipe.


Disadvantages:


It is easier to get lumping if you don't thoroughly whisk the slurry before heating
It must be brought to the boil for at least a couple of minutes to eliminate the raw flour taste, and harder to adjust thickness.
Harder to prepare ahead


Conclusion

Use whichever you are comfortable with.  For fine sauces, roux based may be superior (and certainly more buttery), but you can have excellent outcomes with a slurry.  For casual cooking , I tend to use a slurry, saving roux for more formal dinners and fancier dishes like Thanksgiving gravy.