I'm sorry if our support team doesn't answer your questions.
In any case, if you see the button wobble in Apple's Home app, then the Flic hub has successfully completed its job to deliver the event to Apple Homekit. If you have assigned an action in the Home app and it doesn't trigger whenever the Flic button is pressed (i.e. when you see it wobble in the Home app), then something is failing in Homekit's logic and you should turn to Apple's support.
I suggest you first confirm though that your homepod/iPad/Apple TV is connected to the same Wi-Fi network as your iPhone, Flic hub and the device you want to control.