However, nothing seemed to change afterwards and I was in the same factory reset state where I had logged into the phone and downloaded one app, ROM Manager.

Well, precisely the way as mentioned in the below steps. Note: Samsung's Gear S3 Classic (and Gear S3 Frontier) runs Tizen OS and not Android Wear OS. So, the steps may or may not apply to both OSs.

How to Reset Samsung Galaxy S3 Passcode By George August 3, 2013 April 5, 2022 Forgetting the Samsung Galaxy S3 passcode can be such a pain.

Factory reset all data on a Samsung smartphone Be a mensch and factory reset your Samsung device, especially if you plan to sell your gently used smartphone or give it to a friend or family member.