Hello!
The Jané Muum I see as a very nice, versatile system great primarily for the city, but with wheels and chassis made well enough to endure even some uneven terrains and harshness on the roads. I wouldn't recommend it for brutal terrain, a field without any path, or real forest use... But normal village roads, even those not in perfect shape, will be just fine wit the Muum. You will tackle grass and unevennesses, even if it's not an all-terrain "tank". Overall, the model is truly nice to use and nice to push, so I find it a good choice. Even without inflatable wheels, you'll be fine, don't worry - they are quality wheels with some suspension (even though, don't expect it to rock). The quality materials, chassis and the wheel size is much more important for your ride than just tires themselves or a lot of rocking, even though many people prioritize those features...
If you'd still have second thoughts, and wanted something more all-terrain even (but still not overly heavy), chceck the Easywalker Harvey2 with all-terrain wheels, or something like the Baby Jogger City Premier, the ABC Design Salsa 4 (if you are taller); and from those more expensive models, I would look at the lightweight but very capable Gesslein FX4.