I am not sure by what standard those Colts could be called "service pistols", a term which usually applies to a general issue combat pistol. Most armies, at least in the WWII era either issued or allowed purchase of handguns for personal defense or special purposes, such as military investigators. It is not widely known, but many of those small Colts were issued to doctors and nurses at field hospitals; international agreements allowed medical staff to carry "defensive" weapons to protect themselves and their patients.
Up until this thread, I had never heard the term "socially unacceptable" in regard to handgun caliber, although the concept has been around; it probably prevented most American police from going to the .357 Magnum, and led to the use of .38 Special +P+, a .357 Magnum wolf in .38 Special sheep's clothing.
In any case, I doubt it played a major role in selecting the FN 1903 by any of the nations that used it. One big factor, especially for a smaller nation, is cost, and FN provided well-made weapons at very reasonable cost and was willing to agree to reasonable license agreements, important then as today.