I’m on macOS 12.4 and iOS 15.5 and Universal Control is working for me.
From Apple’s UC web page:
" * Make sure that each device meets the Universal Control system requirements and is using the latest software updates for Mac and latest software updates for iPad."
So maybe macOS 12.4 and iOS 15.4.1 are not in sync? And you need to update to iOS 15.5?
Also just to make sure, you have selected your iPad in the Displays system preferences in the “Add Display” dropdown list:
I’ve found that this selection is not always persistent.