No.. its a structural survey looking at the structure not an abestos survey..I know they couldn't tell me if it was asbestos but in general should highlighting potentialy hazerdous material be part of the scope of a full structural?
Asbestos used to be common in ships and factories with steam boilers, but is very uncommon in domestic houses (except sometimes in outbuildings and garages with asbestos/cement sheets, which are not very hazardous if dealt with by builders or anyone aware of the procedure); so I would not normally expect a home surveyor to be looking for it.