Hi, Christopher from No Man's Land! From one dad to another, I'm regrettably going to give you a very lawyer-like answer: it depends.
There are a couple variables that I look at when determining if the dining plan makes sense or if paying out of pocket is better suited to a family's needs.
From the looks of it, I'm guessing you'll have three people age 11 or older (in Disney's eyes, that's 3 adults) and three kids who count as kids (ages 3-9.) Kids under 3 eat free. The dining plan is pretty affordable when it comes to cost per day per kid. It's also cost effective if you have a kid under age 3 because they can eat guilt-free off of your plate. You can maximize this by dining at a lot of buffets.
The adult price is cost effective if the following two variables are satisfied: (1) you plan to consistently eat a lot each day and (2) you are able to get reservations at the restaurants you want, and in doing so, you plan to have a nice sit down meal each day. Around holidays (what Disney considers "peak season," the Dining Plan price per day goes up a couple dollars, as do out of pocket costs at select table servicing dining experiences.
The peace of mind in paying for your dining prior to arrival (and not having to worry about this expense while you're there,) is especially nice.
Chew on this information and feel free to
follow up with me if you'd like to get more specific or have any other questions.