First of all, if you are not getting free dining then you are paying OOP whether it is for the DDP or day to day on meals. That being said I believe most people paying "OOP" do look at prices and sometimes are

. If dining is an important part of your vacation then 13 days is a long time to be worried about prices and telling the kids "no".
To get the most value from DDP you must do your TS for dinner. At least though you have 1 child and you will definitely save money on that as 1 dinner pays for the entire day. Calculate the entire cost of DDP versus what you think it will cost you for your TS, CS and snack per day.
We are looking at the QS plan for July but my kids are considered adults (14, 11)
so I am on the fence about that as they eliminated one snack per day so the value is even less. Have a great trip.
The problem with that logic -- You're assuming that the OP wants 1 TS, 1 CS and 1 snack per day -- a lot of people don't want that formula.
And I don't get the price shock argument --- The TIW card will almost always work out cheaper than the DDP ---
So yes, paying OOP either way ---
The dining plan for 3 adults and 1 child is about $170 per day....
Let's take as an example --
Some above-average priced meals, so nobody can say I am skewing the numbers cheaply--
Let's assume 4 snacks at $3.50 apiece-- That's $14 for snacks (assuming they want 4 snacks!)
Columbia Harbor House for lunch -- Entrees range in price from $8.19 to $10.19 --- So sticking with above average, let's assume an entree price of $9.50.
9.50 X 3 = 28.50. Plus a kids meal -- 5.50 == $34. Plus 3 adult beverages (though that assume they each want their own beverage) == each beverage is between $1.49 and $2.89 -- sticking to our above-average motif -- I'll assume $2.75 per beverage. We are up to 42.25. They offer 3 choices of dessert -- $1.99, 3.59 and 3.59 -- While many people may actually prefer to skip dessert, I'll include all 1 of each-- $51.42, plus tax = $54.76
Now, lets pick an above-average priced place for dinner -- not the most expensive, but above average-- Let's go with Kona Cafe.
Let's assume the 3 adults get the 3 most expensive things on the menu--
They have a prix fixe-- App, entree and dessert for $29.99 -- So 1 adult can get that.
1 adult gets the New York Strip -- the next most expensive item, at $28.99.
1 adult gets the next most expensive item -- Ribeye -- $27.99.
Child gets full meal for $8.59.
Assuming the 2 adults not on the prix fix will share 1 dessert -- The most expensive dessert is $6.49.
Add 3 beverage -- only non-alcoholic covered under the DDP-- It's $3 per soda, so let's say $9.
Total for dinner -- With tax -- $118.26
Now, under the DDP -- They would have paid about $170, and gotten in value -- $187 --- And that was getting above average priced items, at above-average priced restaurants. Only skipped 1 dessert, but actually got an extra included app by ordering the prix fixe menu. Got the 3 most expensive items in the restaurant!
A total savings under the DDP -- of 9%. And it easily could have been much much less.
Now, let's apply the same meals to the TIW card -- snacks and CS would be the same. But 20% discount on dinner -- So dinner would have been $94.60.
So the total using the TIW card -- Would be $163-- A 13% savings!
So getting above-average priced CS meals and snacks -- and getting the most expensive dishes in the restaurant -- The TIW saved more than the DDP.
And of course -- if you got cheaper meals, the TIW would STILL save money, while the DDP would lose money.
So really, the question is -- For 13 nights -- Do you want to prepay approximately $2200, in order to save *possibly* save up to 10%--- and quite easily actually lose money... (If ordered the 3 cheapest items on the menu instead of the 3 most expensive, then would have lost tons of money on the DDP)
Or only prepay $75-$100 total (the cost of the TIW card), and save up to 15% or more? (if you want to order apps-- still save. Want a beer instead of a soda -- still save).