Sidewalks? Why?

I've ridden loose gravel paths on a road bike with 23mm tires. The road in the earlier picture looks fairly easy but loose in spots.

You will likely be happier on rough roads with wider tires. Especially, given your weight.

Note that the only real suspension on a bike is the tires and your legs (the less you sit on the seat, the better the shock absorption will be).

What about a cyclocross bike? That would be nearly as fast as a regular road bike but gives the option of wider tires.