I would say that on average the route from Grandstand to Springs is five minutes. And that of course depends on how long it takes to load and unload other guests and if there are any scooters involved. But five minutes is a safe bet. The Grandstand is a very nice section, very close to the Springs, it has a very fun pool with spray cannons your kids would love (and its own pool bar and drink station). Another section I really like is The Paddock. Its number 3 on the stop, is very quiet, has a nice quiet pool and playground right there and is between 5-7 minutes to the Springs across a bridge/pond. I love that section. My daughter loved walking across that bridge and we had my mom with us that trip, and even though she was quite ill, she did fine with the walk as well. That section is very quiet which is nice but didn't feel too remote or cut off. It doesn't get a lot of mention but to me, its a great section to be in!
We were also just in the Congress Park Section also in the first building, closest to the Springs and it took us around 5-7 minutes to get to the main area. I loved that section as well. One of the things that made it so nice was that if we dined at other resorts, we could easily walk to DTD and then hop whatever resort bus we needed. The quiet pool in Congress Park is amazing at night in particular as it has a lovely view of DTD.
SSR is very large, however, once you get the lay of it, its easy to navigate around. My husband and I took lots of walks around the resort and because I am a disboard geek, I timed myself several times on how long it took from the various sections to the main area. Even from the Carousel (furthest out) it never took longer than ten minutes at a reasonable pace. I have stayed all over WDW and most of the resorts have long walks involved one way or the other. I do agree, if you have a car,, SSR is even more convenient as you can park outside your door. However, we didn't have a car this trip and never missed it. And one thing about it, unlike other WDW resorts that have only one bus stop, at SSR, your unit is not far from your stop which after a long day is really nice.
I hope you have a great stay!