A patch led by the forum reports: controllers on Steam Deck, the naming screen, and coins in the ceiling. Thanks to everyone who posted.
Controllers and Steam Deck
The store page now declares full controller support, and the game no longer claims to use the Steam Input API — it never did. Together those two were what handed Deck players the keyboard-and-mouse template, so the pad did nothing at all.
Naming a helper works on a controller. The screen asked you to type and refused an empty name, and B and Start did nothing, so a pad player was stuck there with no way out but closing the game. Steam's on-screen keyboard comes up on its own now, and B keeps the default name and plays on.
The same keyboard serves the lobby name and the room code, so co-op can be set up without touching a keyboard.
Touching the screen picks what you touched. Menu presses were aimed at wherever the cursor had last moved, so a tap on a Deck hit the wrong entry, or nothing.
Endless
Coins were scattered at the toy shop's mezzanine height. A tower storey is shorter than that, so a run opened with its starting coins hanging under the ceiling, out of reach — and from the second floor up no coin could be placed anywhere at all. Coins lie on the deck you are standing on.