Short answer = “no.  
Highway=pedestrian is not appropriate in this setting
The operator's map describes these as "main paths".
The same operators map makes a feature of "Downhill Walk" from 100 m asl high Cable Car terminus (south centre of the Gardens) to Tree House then northeast towards the CBD.  This 2 km walk almost exclusively uses these very broad ways (as segments are called in the operators map).
The highway=service road tag lets OSM mirror the operator’s map.
Roads tagged here in this way are typically asphalt paved and about 3 metres wide.
There are many paths that service vehicles can access in the normal course and which I have left as highway=path.
OSM requires contributors to first mark what is there (in this case a road) and then mark who/what has access. 
In this case, tags such as "Track/Land access" might be an alternative when supported by tags that make such a highway visible.
In this case, tags such as "Path" are considered unhelpful as the operators “main paths” will be “lost” amongst all the other paths, enjoyable as they are in their own way.
