Yes, I agree that if the person is already registered that should be sufficient.
However that is not how the current system operates. Neither will the proposed opt-out system. Of-course it would be useful to understand exactly why that is, but there must be good reasons, even if only to maintain the trust of the public in medical staff and procedures. It would also be useful to understand why, actually, consent is not given - I think this is in around 50 - 60% of the cases (from previous links)- at the lower end in the opt-out system.
It is true that there are plenty of potentially donated organs that are unused, but this is not because there is an excess of donors, but because most organs cannot be used - depending on the circumstances of the death, matches, age of the donor, medical history and so on.
In fact, very few available organs can be used, so there is a desperate shortage. Imo, that is why the issues need to be addressed openly and transparently.