The Switch its self does not know how to get to any other subnet than 192.168.0.0/24 ,
does not know how to return the traffic, that's why to need a route for to manage the switch remotely ( helps to always think about both directions separately )
You will need the route to access the management of the switch device remotely
you not need it for clients on that network to operate successfully.
asside: ipforwarding does not need to be enabled