Can I install and manage elemental derivatives purely with elemental toolkit cli and without the operator?

Sure it can be done, but it also depends on your expectations of what

manage elemental derivatives

means to you. I assume you are aiming for a Rancher less setup (otherwise I see no reason not to use the operator). In that case I'd say it is handy that your derivative already ships provisioned with k8s and all the software you expect to be running. The operator provides an integration layer with Rancher, so that the nodes can be easily registered and provisioned within Rancher using Elemental images. I'd suggest you to take a look to the Elemental example images as they do not assume or provided anything related with the operator.