I played the game entirely through the main Steam Big Picture gaming mode rather than desktop mode. I used Desktop mode exclusively for copying the files to the Steam Deck and adding it as a Non-Steam game.
It's possible to get the weird non-pausing behavior as early as the very first room of the game by rapidly rotating the stick and mashing start a lot. When this is happening, the pause sound can be heard, but the menu doesn't display / doesn't close.
https://www.twitch.tv/annmwhite/clip/SpineyAnnoyingCroissantNononoCat-JxJXU3ecbL...