I had the exact same problem, and i tried alot of tricks and tips, and nothing worked for me 'till i tried something i found on another thread " https://answers.microsoft.com/en-us/msteams/forum/all/show-background-effects-option-has-disappeared-on/1e0490ce-3a6c-45f0-89a5-0d8c615c5d31?page=4 " the solution came from JoeWillis44
A similar work around to this issue to disabling the camera, perhaps a little quicker.
When initially the device settings were for the default PC Mic and Speakers, there was no background effects button/options available.
I had seen somewhere that if you switched to custom settings it would show up, I had no other choices on this device so couldn't make a custom settings device profile.
I plugged in a pair of USB Headset/Mic earphones and then switched the audio devices to that, Poof! there is the button ready to go, I unplugged it, the settings went back to PC Mic and Speakers but the background effects option remained and seems to be working.
There must be something that is preventing this from being enabled initially, and its weird because there is no rhyme or reason to it, I had one user unable to do this on a device, then another logged in and it worked fine.
Hopefully this helps someone.