I think that FMS is a great measure of general physical capability and teases out imbalances very well.  I also think it is a great way to take: 

1) someone who is new to CF or new to fitness and take them to a level of physical readiness. 

2) take someone who was injured and rehab them back to a level of readiness. 

3) balance out an athlete, either new or experienced, who discovers they have a lot of imbalances between their left and right sides.

So in those applications, FMS gives you a very systematized way of doing that, and also not as a clinician but as a coach.  Their roadmap is easy to follow and is excellent and easy to implement.  FMS also helps you prioritize which problem to fix first so that helps as well.  It turns out that the correctives may also magically address other issues you found during the screen, while fixing the first problem you started with!

I believe FMS has been in development a lot longer than MWOD has been around and IMHO therefore is more complete.  MWOD’s mobility techniques are more numerous than as described in FMS.  That doesn’t mean that FMS practitioners don’t know MWOD mobility techniques. Many already do. It’s just nice that Kstarr took them public with the help of the internet.

But IMHO I think that MWOD’s current content and messages are missing the stability pieces of athlete preparedness. It is true that if you mobilize, you can often get an athlete’s body to magically turn on correctly. However, there are many cases that after mobilization, the athlete still really isn’t ready to build fitness and strength on top of dysfunction, as Gray Cook likes to put it. And mobilizing more isn’t going to help or be optimal in the achieving the end goal.  FMS provides a systematic way to do that.

So here is where FMS can be very additive to MWOD. You determine if FMS is necessary or desired. You start with mobility using FMS as a guide and enhance with MWOD techniques. You then move to static stability with FMS correctives, and then onwards to dynamic stability. then the athlete is (more) prepared to do whatever they want to do.

Where MWOD is more proficient is preparing you for Olympic lifting ala Category 1 thru 3 movements.  FMS IMHO is more a general preparedness whereas a lot of MWOD is gearing towards CF and Olympic lifting preparedness.

In answer to your example, FMS’s deep squat (and its other tests) test to the extremes of an athlete’s potential to really make imbalances and problems glaring.  so FMS’s deep squat has your feet at shoulder width and arms at a specific width as setup by a bar held overhead. But just because you can’t do a good FMS Deep Squat doesn’t mean you can’t do an overhead press and achieve the deep squat position.  For example, I can now get to deep squat position and with a bar overhead, but my arms need to be way wide and my stance is slightly wider than shoulder width – this still means i would not get a 3 on the FMS since the FMS is specific on testing setup. 

Thus there could be many cases where MWOD test and retests of the archetypal shapes would be more valid than using FMS.  But to give a counter example, just because I use MWOD techniques and now can achieve an overhead arms position without movement in my ribcage, doesn’t mean i have enough stability in my shoulders to suspend any kind of weight overhead safely, or in a dynamic situation.  You’d have to test and build up for that after you achieve the position through mobilizing.