Was there any logic behind this practice (rubbing frostbite with snow)? Is this method encouraged /
discouraged by modern medicine?
According to Hypothermia, Frostbite and Other Cold Injuries: Prevention, Survival, Rescue ...
By Gordon G. Giesbrecht, James A. Wilkerson, the treatment gained popularity in the Napoleonic Wars because
rapid rewarming from open campfires or other sources of dry heat
caused so much devastation.....Dry heat from ....open fires....cannot
be controlled. Excessively high temperatures are usually produced,
resulting in a combined burn and frostbite, a devasting injury that
leads to far greater tissue loss.
This treatment is not advocated by modern medicine. According to Survive Outdoors, Inc:
Until the 1950’s frostbite treatment consisted of rubbing snow over
the affected area. However in 1956, Merryman disproved this treatment
intervention. He instead encouraged the public health service medical
officer in Tanana, Alaska to try rapid re-warming, with great
success--Hence, the beginning of a new and effective treatment for
frostbite.
Edit in response to comment from @imsodin, who said "rapid re-warming could be misinterpreted". True. The Mayo Clinic site says:
Gently rewarm frostbitten areas. Soak hands or feet in warm water — 99
to 108 F (37 to 42 C) — for 15 to 30 minutes. If a thermometer isn't
available, test the water by placing an uninjured hand or elbow in it
— it should feel very warm, not hot.
The Mayo Clinic site also says:
Don't rub the affected area and never rub snow on frostbitten skin.
Thus, not only rubbing with snow, but rubbing with anything, is a no-no for frostbite. Rubbing will further damage already damaged tissues.
Caveat: This answer does not attempt to describe the proper treatment of frostbite; for that, see What are the different stages of frostbite?. The excerpt from the Mayo Clinic addresses only the proper temperature for rewarming; the entire process is more complicated and rewarming in the field is not always the best course.