What category room was that at the Poly that came back $7000 for a week long package?
First week of December is Value season (LUCKY YOU!). I know you said the dining plan saves you money, but it forces you into a package. You can almost ALWAYS get a "room only" discount of up to 40% off, especially that time of year. Skip the dining plan. Only buy park tickets you will use (like, if you won't hop, don't get the hopper option. Don't get the "no expiration" option. Don't bother with the water parks option in December. Weather can be iffy.
I just priced out a trip first week of December for a family your make up. Price was $5105.65. This was a standard (Garden) view room. You have to de-select the hopper and water parks option, as they include that automatically. Once you get it down to 7 day Base tickets, this price INCLUDES the dining plan.
Take away the dining plan and tickets (making it room only), and your room rate is $365.00 per night. This is rack rate. Total for room only would be $2874. Now, assume that Disney will offer a 40% room only public discount (like they did the past few years for this week). Your NEW room total would be $1724. Tickets will run approx $1300-1500. You can buy the tickets once you get there. Room only has a more flexible payment/deposit policy. You only pay one night upon the reservation. The rest is due when you check in, so you can have some extra time to save some money before you have to pay it. As mentioned, you can buy tickets there. Bring easy food for breakfast in the room. Then have CS meals for the most part, throwing in a few TS meals during the week. That's what we always do. We save a lot of money that way.
We are staying at the Beach Club, in a Club Level room the week BEFORE Easter. We were able to take advantage of the 40% military discount. Our room rate is $381/night, and this is during PEAK SEASON. Our room total is exactly $3000. We bought tickets for $1200 for the four of us. We are not doing dining plan, but will probably spend around $1000-1500 over the week on food (we have some signature dining planned). Our total spent should be around $5500, excluding airfare. This seems EXTREMELY reasonable to us, considering we are going over spring break.
You can do this rather inexpensively. Keep in mind, the Poly tends to be very pricey. You can stay at YC/BC, or BWI for less. These resorts tend to have more rooms offered at a discount. If you can score a Garden Wing room at the Contemporary, that's even cheaper!
As for "how do people afford it". Simple: Some people just make more money than others, and prioritize their lives differently. For us, I am a SAHM. DH is in the military. We are far from rich, but we live a very modest lifestyle. Vacation time together is priceless for us, as DH is away from home about half the time. My DH said to me the other day "Money is no object on this vacation." And, this is coming from a major cheapskate! This vacation will come about two weeks before DH leaves for a 7 month deployment to the middle east. So, for us, it's not about cost.