I finally managed to use my TTP card at the gate, still puzzled over the technical issues. HND T3 has 4 gates, of which the 2 on the outside are for Japanese and the 2 in the middle for foreigners. I believe last time I entered it was only one for foreigners, the left. The one on the right did read my passport and card, even processed me through all the steps, only to tell me that they couldn't confirm my card data or something and fail. Does that mean the problem is with the card or could this happen if they can't match the fingerprints?
Tried the other one, which immediately refused my card. Same experience I had last time. The staff responsible for the gates seems to be the same as the next immigration entrance left to the automated gates, so I went over and one of them guided me to use the right side again - which worked this time. Only thing different was that I used the fingerprint helper they had on the gate.