Whether you need sus depend on what you're riding over and how fast you're riding over it
It's certainly been good for us, allowing fast fireroad riding, including the usual rocks and pot holes whilst not even waking the little one. We started at 5 months, with an extra baby support harness in there too. Tyre size is important too, if you've got 20" wheels then there's a wide whoice of fat BMX rubber to choose from.
As Tom said, Chariot with a third wheel is fantastic for walking and running with. We've used it loads over very rough ground, for getting around ski resorts - the only restriction is the width really. Amazing maneuverable for something that looks so big.
Obviously if the "cockpit" is really comfy for the child you can get away with rougher stuff. And comfy cockpit and good harness systems can indeed make the occasional (ahem) roll barely noticeable