I've tried it with my oppo reno, my partner's Samsung galaxy A70 and an old OnePlus one - case on and case off on all of them, two made of glass and one is metal/some sort of rough finish.
I've used all of these phones with previous quads I have owned including - bebop 1, DJI spark, 3dr solo, Upair one plus, hubsan zino pro and several others and never had a problem with my phone been held in place.
Based on your evidence parrot should market they're sky controller 3/anafi as only been compatible with iPhones.
All phones have a generic rectangular shape so it's not rocket science to design a phone holding mechanism which is suitable for all.
If you think that it's only people with plastic android phones which have this problem you should watch this video as it cost him $400 to replace his iPhone screen when it popped out of the skycontroller 3 phone mount and smashed on the floor.
The flaws in the design I can see include
. The rubber grip pad is too hard so doesn't seem to create a great deal of friction - softer rubber pad may help.
. The area which actually comes in contact with your phone is very small, could do with 2-3 contact points top&bottom, obviously controller design doesn't allow that.
. The arm is angled leaving a gap between the back of the phone and the arm - could do with been flush.
. The arm when extended has movement allowing the phone to move closer to the edge of the grip pad and closer to slipping off.