This looks like the idea EPRI floted years ago, but to my knoledge I don't know of any takers.
The problem with leaving the neutral open or high impedance grounded is it reduces the ability to flow ground fault currents, and thus the ability to detect a ground fault on the line. And because ground faults can transition into phase to phase faults we do want to clear them fast.
The transmission lines are not the only inducable conductor. In most of these cases the earth also acts like an inducable conductor, which is why the DC potential in the earth causes current in some lines.
Also note there can also be current flow in shield wires, pipelines, etc, although they go un-noticed because there are no seen effects.