I third the trails off the lower Livermore Road. While short and without much elevation gain, some of the trails are rough in places. You can easily push an off-road stroller up the Livermore Road.
The trip to Greeley Ponds doesn't have any rough sections though it's a bit longer. No need to drive around to the Kanc. There are several side trails off the Greeley Pond trail as well, like the Flume trail. I've only been up it in winter so I'm not sure what the footing is like.
The X-C ski trails should also make for some mellow strolls.
-dave-