In addition to @Andrea_Mayer’s good post, and as you suspect I think, you’d need to just create a project for this purpose, perhaps call it Task Templates, make it accessible to all, and house all your task templates there, but they’d need to be re-homed (add to project) to the destination project.
You could potentially automate part of this, as I describe in this post:
@Forum-team, the help page reference is a doc error if you can pass along.