On our arrival day, adrenaline takes hold and none of us can nap!
Every day after though, naps are our #1 key to keeping everybody happy! We get the kids (age 4/6 last time, they'll be 5/7 for our upcoming trip) up at their usual wake-up time, around 7am. We hit a park at rope drop, have lunch just before noon and head back to the resort for at least 3 hours. All four of us take a nice, hour-long nap (something none of us does at home), and sometimes a swim, and then we go back to the parks around 4:00pm and stay until 8:30am (the kids' usual bedtime).
In our experience, between the naps, not spending more than 4-5 hours "touring" at a time, and not keeping the kids up late (except for the last night, when anything goes!), we've never had a single meltdown, and the kids haven't needed strollers at any point. It's amazing how much you can see and do in the parks with just 8 hours of touring, if you travel during a non-holiday season and plan your day wisely. (Not only that, but if you're napping between 12pm and 4pm, you're conveniently side-stepping the hottest, busiest, most crowded and most potentially miserable time to be in a park, and arriving back just as the crowds begin to lessen...) Last time, we had the equivalent of 5 full days in the parks, and were able to see and do almost everything (not to mention do our favorite attractions multiple times). We'll have 6 full days next time, which will give us the flexibility to relax the pace even more.
So my recommendation would be - embrace the nap! You may be surprised at how much you and other adults in your party need a nap, too (or just a chance to lie down and rest in air-conditioned comfort). The average WDW guest walks 10 miles a day...