It looks as if the footpaths in the centre form a routing island. Graphhopper (via the OSM website doesnt use them at all, but OSRM will use the paths & even routes round the edge of the car park). The tagging looks accurate albeit unconventional (dont know many places in UK were snowmobile access is a thing). It could be that routers throw islands away. 

Probably the simplest thing would be to connect the two paths at the car park to the service road in the car park.