@sud, BTT only shows the privacy window it it doesn't have access to the Accessibility API. This would also explain your gestures not working (most of BTT's actions require that API).
It is possible your privacy database is corrupted, this unfortunately sometimes happens due to a macOS bug.
You can reset the database by running this terminal command. Before running this make sure you don't have any unsaved work, also quit BTT before running this.
sudo tccutil reset Accessibility
After that you will need to add all apps to the Accessibility section in System Settings => Privacy & Security again. However this often fixes permission related issues.