Brine, then marinate

There are several reasons why you should marinate before cooking:

• Many marinades contain raw ingredients that should be cooked along with the food being marinated, such as garlic or ginger. In some cases this may actually be a health hazard (raw garlic can harbor botulism), in other cases you'll simply end up with an undesirable pungent flavour.

• Many marinades also use some amount of acid, which helps to kick-start the process of breaking down the connective tissue and tenderizing the meat. This is an especially big deal for chicken breast meat and tougher cuts of beef or pork, because most cooking methods (except for slow-cooking) don't do much of this and tend to produce a tougher, chewier end result. Once the meat has already been cooked, using an acid-based marinade won't give you much more tenderization; you need the combination of acid and heat.

• Finally, many spices release most of their flavour when heated (cooked). Examples are cinnamon, saffron, star anise, and various types of hot pepper or chili powder. If you marinate after cooking, you won't get much flavour from these.

Of course if you are using a very simple marinade such as soy sauce and are marinating only for flavour purposes, then it might not matter much, but that's the exception rather than the rule. Marinating afterward can certainly improve the taste, but most of the time you'll end up with better results by marinating beforehand.

A few things aren't quite right here.

1- a cooler full of water for 3 days.
A turkey will take 3-4 days to thaw in the fridge- between 35-40F. Quicker methods call for submerging in running water for some hours. Submerging the turkey in stagnant water- even if it started as ice water- will allow the turkey to rise well above 40F over the course of 3 days. You would have to add ice or have the turkey somewhere colder than 40F for this to be safe.

2- brining too long turning a turkey to mush.
The salt in a brine denatures some proteins and supercharges the bird with water. These will make the meat more tender and seem more tender respectively. That said- the risk isn't that the meat will get mushy. Unless you are adding some other ingredient to the brine that is actually a proteinase like papain the risk is not mushiness but being too salty.

3- 1-2 days of brining.
I have never seen a recipe that called for brining this long. The brine recipes I have seen are on the order of 4-10 hours. If your brine recipe calls for this length of time then it is probably more dilute than the ones I use. In such a case it would probably be fine to thaw it in the brine if you address the safety concern.

The turkey being frozen will prevent the brine from penetrating but as the bird thaws from the outside in the outside of the bird will have more alone time with the brine than the deeper meat. I can't say whether this would be a problem- especially as Harold McGee says that the salt from brining isn't able to penetrate very far into the meat anyway.

Unless it is cold enough outside (or you have enough fridge space) that you could brine the turkey at <40F for 2 days while it thawed I would recommend at least mostly thawing it before introducing it to your brine.