Is it possible to configure a "Job Template" (this is a particular kind of Jenkins item) with the groovy template coming from SCM as is possible in other job types? I don't see it as an option in ours, but perhaps there is another plugin required for this?
There is not such an option. The closest equivalent would be to create a job template using the Pipeline transformer, where the inline script is merely boilerplate running a load step to load the “real” code. (An existing RFE CJP-1718 suggests removing the need for this boilerplate; if that would be useful to you, file a support ticket mentioning it.)