Hello samppa10s, I appreciate you providing a video, I have had a good look at this and looked at the settings you showed us and couldn't see anything that stood out on why you are unable to use the chat option in game.
I'm interest to know, if you change your squad privacy setting to something other than just 'Friends' in the menu, are you able to use the chat option then?
Can I also ask, have you been able to try all game modes and does the option for text chat not show up for any?
For the sake of being thorough, can I just ask, have you been able to try this on another controller to see if the text chat works then? have you also tried connecting the controller via usb to test this and also finally resetting the controller using a pin on the back reset button? I need to be sure that this is a game issue and not a unusual peripheral issue, so if you could test this that would be grand.
Finally can I ask what Playstation firmware/ update are you currently running on your PS4? I can see your game version for Siege is currently 15317359/