We rarely rent a car. Now our kids are older, but after many trips to WDW without a car I would have to say you'll be fine without one. Midday breaks are a must for us and you'll find that more often than not, the bus will be deserted when you return for your naps! At night the busses are crowded but then again, so are the trams. And getting to a broiling hot car that's been sitting in a parking lot in Flordia for hours is not much fun either!
My best advice is don't rent a car and take smaller strollers. I can understand the beauty of a large stroller (I loved mine) but they are a pain to fold up and get on the bus, whereas a small one, not so much.
I also dine at the resorts and can say that while you do have an extra step, its not as bad as it sounds. Just get on the first bus that comes along to any park, exit that bus and then simply walk over to the resort bus you need. Very easy, just allow the time you need to do this. If you dine at an MK resort, that's even easier as you just hop the MK bus then walk over to catch the monorail. For sure renting a car is nice for resort dining, but bear in mind parking lots can be a hike to the main area you want to be in for your meal. Once you leave your meal to go to a park, you then have to repark the car at the park, then get on a tram to get inside the park, vs. hopping on the bus.
There's inconvenience attached to any mode of transporation at WDW. Having a car is great for grocery stops, side trips and resort to resort transporatation. But it has its down sides too! You just have to weigh the pros and cons to both sides!
Knowing our luck that would happen EVERY time. 
