Hello, everybody. I have solved the problem in the Genius Bar of the Apple Store where I bought the iPhone.
The bad news is that it is a hardware problem. The NFC antenna deteriorates and ceases working, which affects the NFC reader and can affect also Apple Pay (which is the other "side"). It was my case, but not sure if both things happen always together or not; some report only problems with the reader and not with Apple Pay. The issue is not detected by the hardware tests of Apple. In fact, what they have done is to test Apple pay with a NFC reader and see it doesn't work. The really bad news is that all the antennas are a part of the motherboard and the only repair consists in replacing it entirely, which costs dearly. In Spain, around 400 €, or $500.
Of course, come to this point, I would have considered buying an iPhone SE (mine was an 8), which is only 100 € more or so. But the good news for me is that it was covered by the guarantee, because here in Europe it is 2 years. So I have gone out of the shop with a shiny, mint, brand new iPhone 8. I have restored it from the last backup and the NFC reader works perfectly with LibreLink. I had the latest iOS update (14.7.1), so it seems to me that the claims of lack of update of the Abbot App are not justified.