I agree with this part of your statement James.
Wrestlers don’t have to run, but it doesn’t mean that it is a bad option. Our university has their wrestlers run all the time and I don’t see any problem with what they are doing. I don’t work directly with the program, but they have produced a number of Olympic gold medallists and numerous other medals. I do see some issues with the volume of work they do on the mat (leading to lots of soft-tissue injuries), but running is not the problem.
Problems I’ve seen with bikes, ellipticals and rowing machines is that they can lead to other over-use problems. Wrestlers have chronically tight hip flexors due to the position they are in most of the time (hunched over). Upright running at easy intensities and on soft surfaces can help to open up the hip flexors in a restorative manner. Stationary bikes and rowers can exacerbate the shortened hip flexor issue, and I’ve also seen lots of hip and knee issues with steppers and ellipticals that do not provide lateral floating (i.e. the foot does not track to the natural center line like in running. You are stuck in the track of the elliptical or stepper).
The pool has also been a good environment for mixing up tempo work to alleviate overuse issues. I think it is important to work different options into your higher volume, low intensity work. It is important to note that the term “tempo” work does not necessarily bind you to running. Charlie has discussed lots of options in the pool.
I agree with Number 2 with everything above, when I was in University I did tempo work on treadmills and stationary bikes: The result was that my IT band had tightened up and the middle/outside quad muscles were overpowering the inside quad muscles. This lead to the patella tracking to the outside and causing and grinding the patellar tendon. I needed rehab and time away from the track.