An interesting experience: I did "cold bath" after running on only right calf in a period because of pain in the achilles tendon (no pain in the left ). The calf felt much better than the other after a while. It was very clear to me towards the end of a couple of marathons: The right calf still felt good and the left was tight and struggled with cramping in that calf.
When I think about it afterwards. It is a perfectly design experiment in terms of exactly the same conditions for two legs except for cold bath after running.
So I think cooling is beneficial for recovery under certain condintions. But also listen to other experiences and I think there is no contraindication that cooling also can lower the effect of training that you want. I mean if you reduce training you could also feel less pain and stiffness in the muscles depending on many thing. The recovery and training is indeed complex and you should be critical.