I only know the situation in Germany, but it’s likely similar. Consent is not the bottleneck. The number of organs that cannot be donated because consent is either unclear or not given at all is pretty small, even though a lot of people aren’t registered organ donors. Usually the next of kin are asked what the donor might have wanted and in the majority of cases, they agree with a donation.
Making donations mandatory would have a very minor effect.
The true bottleneck is that a very specific condition is needed in order to make organ donation possible. The donor has to be dead because you can’t take vital organs from a living person. But at the same time, their heart still has to be beating in order to keep the organs supplied with oxygen until they can be removed.
The only way this is possible is when the brain is dead but the heart and other organs still function with medical assistance. A person without a working brain is considered dead by law. This only happens with very serious brain injury, major strokes or similar incidents. And with improving treatment options, a lot of people who would have ended up brain dead a few decades ago, can now survive with varying neurological outcomes.
If you want a lot more organ donors, ban helmets. Otherwise, there are no simple solutions.