Control Center add button not working

Okay. There shouldn't be any settings that prevent you from adding buttons to the Control Center, but I thought you should try resetting your settings anyway and see if that makes a difference. This will mean that you have to set up your iPhone's settings again. No content is lost with this.

Source, Apple Support:

Reset All Settings: All settings—including network settings, the keyboard dictionary, location settings, privacy settings, and Apple Pay cards—are removed or reset to their defaults. No data or media are deleted.

Go to:

Settings → General → Transfer or Reset iPhone → Reset → Reset All Settings

You can find more useful information here:

Return iPhone settings to their defaults - Apple Support

All the buttons can be added. Just tested it. There is no function that can disable button additions inside Settings → Control Center, so it sounds like a glitch.

Try the "universal trick" - try to do a Forced Restart of your iPhone:

1. Press and quickly release Volume Up

2. Press and quickly release Volume Down

3. Press and hold the Side Button, the phone will restart. Release when you see the Apple Logo.

Once the phone is restarted. Try again. Also try removing all the buttons in the Control Center and try adding them again.

Scroll Down to the bottom and tap edit.

try resetting it lol