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.
I’m not sure about the mountain buggy rubber ones. We do a lot of walking over different terrains. We had a different buggy with rubber wheels and it just wasn’t cutting it. We got the mountain buggy but with the inflatable tyres about 7/8 months ago now. Obviously they are able to get punctures. But baring in mind we use it 2-3 times a day for muddy/bumpy/sandy walks, it’s only punctured twice and it’s very easily and cheaply fixable !
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.