Roofs are very much subject to Owner preferences and prejudices.
I would check out what kind of roofs the Owner already has. If there is a similar structure, ask how they feel about maintaining another one of those roofs.
if your structure can hold it and it doesn't blow your cost, you could put a topping slab above a waterproofing layer, which would open the door to a whole bunch more products (sheet waterproofing, elastomeric, crystalline, etc...), and give you design flexibility for the recreational use. Also, you wouldn't get the UV damage. If you do this approach, check that the topping slab won't move. insulation could be over the waterproofing or on the interior, constraints being condensation possibility & space for interior and drainage & long-term structural integrity for exterior approach.