Man 1 applies ice to trained body part for 20 minutes following all workouts for a year.
Man 2 does not.
All other factors equal, who makes more progress in that year?
Doubt if there is going to be any difference, cause whether you ice or not, the amount of recovery you will have from a workout will be the same regardless - the ice will only accelerate the rate at which you recover. And I doubt if 20 - 30 minutes of icing once a day is going to make you recover a whole day or two sooner than you otherwise would.
Of course, if you have an injury and you ice every few hours as part of the treatment, that will make a big difference in how much you recuperate.
There could be more factors involved - not sure if there is more than the above reason for the sporting organizations to make their athletes do the ice bath thing.