Apple - No option to remove Bluetooth device in iOS Settings app?

Workplaces and schools can prevent bluetooth modifications via an installed configuration profile. Check for the presence of such a profile at Settings > General > Profiles.

Using Apple Configurator 2 (an application made by Apple to allow companies & schools to administrate their employee's/student's devices) there is an option:

Allow Modifying Bluetooth Settings (supervised only)

Unchecking this option in the installed configuration profile will prevent iOS from forgetting the device.

Note: It is the same for Profile Manager in macOS Server

If you are using iOS with an installed configuration profile, I would suggest contacting your IT department and requesting that they allow you to edit Bluetooth settings.

Heres a picture of what it looks like in Apple Configurator 2:

I've experienced this phenomena as well, usually have restoring from a backup, and found that I had to delete the apps associated with the devices in order to make things work properly.