I don't think any legit production outside of Ed Wood movies or other horrendously low-budget movies from the early eras of film used "models on strings."
The Romulan ship in Balance Of Terror was hung from wires.
This is the first I've heard that. It's possible, but are we sure? They did all the other model that I know of work on pylons.
John Peel's Star Trek Files magazine issue called The Star Trek That Almost Was ((c) 1985 Psi Fi Movie Press Inc) has a number of "behind the scenes" photo's as illustrations. On page 20 (where he discusses John Meredith's The Lost Star) there is a photo showing the model suspended from some sort of rig that appears to be rotatable - there are signs of horizontal strings/ropes attached to the rig.
In the same series, the issue called The Star Trek That Never Was (same copyright), the photo on page 30 shows someone holding a clapper board under the model - there is clearly no pole supporting it.
In neither of these photos are strings visible, though they must be there.
Also neither of these photos are identified with any caption, however it is clear what they are.
(as a side note, the blurb on the back of these issues is the wrong way 'round - the cover for The Star Trek That Never was should be attached to The Star Trek That Almost Was and vice versa.)