Okay so i'm having this same problem, what I've found is that the installer downloads the json file with the data but doesn't actually download the launcher, so there's no actual way to run the forge version and i'm completely stumped because i have no idea how to fix this. I thought downloading a copy of the forge version 1.12.2 launcher would work but literally no version of the launcher prior to 1.13.2 exists on the internet and to add the minecraft launcher doesn't know what to do with the json file so it just kinda sits there as a profile with no way to delete it or edit it's version or profile name. so i have no clue i hoped this helped bring more light to the situation.
Edit: i mean the launcher jar file that reads the json for forge 1.12.2 if it didn't make sense.