but since when has any self-respecting scientist been led away from a beautiful hypothesis by pragmatism? Much less a physicist?
More frequently than you might think.
I highly recommend Lisa Randall's "Warped Passages" as a fairly recent meditation from a working physicist on the strengths and weaknesses of multi-dimensional theories, as well as some of the background on what motivates them.
The short story is that higher dimensions make it easier to generate universes like ours, which are almost flat gravitationally and have this ridiculous difference in scale between gravity and everything else. Regular 3+1 dimension Standard Model physics requires a ridiculous level of fine-tuning for this to happen, so it is interesting to look for deeper theories where these features arise naturally.
The problem with String Theories in particular--and as other posters have pointed out it really is a family of theories--is that they are almost useless for "model building", which is the activity of generating simplifications we can actually calculate with to describe the world as it is today vs the world as it was a few Planck times after the Big Bang.
So while string theory is neither "untestable in principle" nor "designed to resist testing" it is "naturally test-resistant." Some simpler variants--including a rather elegant heuristic model that Randall herself co-created--have been killed off by the LHC, but the diversity of possible low-energy models enabled by String Theory is so large and heterogeneous that in practical terms they cannot all be falsified.
In the meantime, String Theories continue to take up an inordinate amount of theoretical physicist's time and effort, and there have been fairly senior voices raised in concern about this, particularly because it has been going on for so long that we may end up with a generation of theorists who have never done anything but string theory and have come to the conclusion that it won't work.
Rather than waiting for this Big Reset, it would be nice if we started spending more time looking at non-stringy alternatives.