I have a Nuna and it’s brilliant for these activities. Mud wise walk through a puddle or grass on the way back and it gets it off. Or Chuck water over them before getting back in the car. I have a picnic blanket turned upside down in the boot for it.
We got an out'N'about buggy with pump up tyre's. It's been an amazing buggy, we go everywhere with it country walk, up the woods down the coast. Had it 18 months now only had 2 punctures. We keep a spare tube, tyre levers and a small pump in one of the pockets on the buggy. We did upgrade the tyres to something with a bit more grip straight away.
You Could message them and see if they still have the original wheels put away somewhere
You don’t want inflatable, get full rubber otherwise you’ll get punctures.