Drawing Models > Replace is indeed the best route to making 'a variant drawing which is the same drawing but of a different instance'.
If you have fifty models, as in the first sentence, the easy way would be to make a mapkey or two to automate the steps that are not unique to each drawing, minimizing your time to providing the name of the new model and selecting the replacement instance.
If you have thousands of models... first you should consider if you really want thousands of drawings, or if you want to make them on an as-needed basis. If you do really want the drawings, then you'll want to automate it more. Probably the easiest form of automation is constructing a trail file -- run through the steps of making one instance's drawing, and look at the trail.txt file made thereby. Replacing the places where the new drawing's name and the instance's name are entered/selected with a different one, you can then have a trail that you can run to produce the next instance drawing, and this replacement process can be automated with tools like 'sed'. You can also repeat the portion of the trail that makes the next drawing, saves, closes the window, and does Erase Not Displayed, and make one trail that will make each of the 1k drawings in turn. Just be sure that you do close windows and erase models, or your session will quickly get too large!
For a more customizable approach, you could employ Toolkit or other programmatic API, but for the case you've got where the drawings are 'as achieved by Drawing Models > Replace', you probably don't need to go into that.
