Back in the day, nearly all Law Enforcement Officers carried revolvers. At the time the 10mm was developed, bullet design was not where it is today. Today's defensive ammo has very reliable penetration and expansion, even after passing through layers such as denim and leather, making the 9mm Luger a viable LEO carry option. Stopping power back then, was considered to be proportional to bullet diameter, since the bullets of the day often did not perform optimally. They would often over expand, or over penetrate. Bigger diameter, more powerful rounds, were felt to be a necessary solution.
The 10mm is essentially a magnum that falls between the 9mm-38 Special-357 mag , and 11mm-44 mag. Law enforcement thought they wanted more power than the 357 mag, and few officers (Clint Eastwood excepted) could shoot the 44 mag well, or wanted to carry a revolver that big and heavy. It turned out that the 10 mm still generated more recoil than many cared for, and the revolvers were still heavy to carry. For that matter, very few LEOs shoot the 357 mag as well as they can shoot a 9mm. As service calibers became larger and generated more recoil, shooting proficiency qualifications fell. This was a problem as no LEO departments wanted to have many or even any, officers who could not qualify with their service arm.
The 357 Sig came later. It approximates a 125 gr 357 mag round and can be chambered in a reasonable sized auto loader handgun.
Revolvers are more difficult to shoot well than auto loaders, without a lot of practice with the revolvers. . Qualification scores improved when departments switched to auto loader handguns in 9mm Luger. Large capacity weapons and the ease of reloading with magazines, compared with speed loaders, was another advantage.
This is not a commentary on caliber or recoil. An interested, determined shooter can learn to shoot a heavy recoiling caliber very well, with a lot of practice and motivation. But LEOs only have a limited time to train, unless they are SWAT, Hostage Rescue, or Seal Team 6. And they need to spend the majority of their time doing police work, rather than remedial training to handle large caliber weapons that they can't shoot well. FWIW, I understand that many LEOs are not shooting enthusiasts and do not practice shooting (for recreation as we do) other than mandated range times and qualification testing.