I did some more testing and I the only issue with controls I could find is not all joypad controls working in browser on steam deck.
I tested the following browsers on desktop and all work without any control issue:
Chrome, Firefox, Vivaldi, Brave, Edge.
Both keyboard and controller work, even my dualsense works wirelessly now.
Note that the buttons are not supposed to be clickable, nor touchable, input only via keyboard or controller. The buttons are interactive during the loading screen, which shows the control mapping, you can test input there. The dog buttons should play a small animation and a sound snippet.
Can you give me more info what is not working on your end so I can look into it?
(Does anyone know why some posts appear both here on the rating page and the game pages, but replies only on either?)