Requiring consent of the donor (or their family) may seem silly, but removing that safeguard would inevitably lead to both abuse of the organ donation system,

I believe that the increase in legal and ethical supply would reduce the amount abuse.

and also a distrust of the system [...]

That's the real problem, but an opt out system would be a good compromise.

Philosophically, it's also important to note that the organs aren't actually harvested from corpses per se, but a heavily sedated person (who may or may not still be showing brain activity). They rule the death before they've harvested the organs, I think, but the person still being alive at the time of harvest is a big deal for organ viability. [...]

The question is how you define personhood, but if you ask me, the body is alive, but the person is dead.

The point here is that it's inherently an ethically dicey proposition, similar to how deciding to switch off someone's life support of a non-donor can be a big decision for families and/or doctors.

No, in that case the person could be alive, maybe even conscious, but unable to interact with the world ever again.

In a way, the consent requirement can be seen as a way of sidestepping the messy philosophical questions like "what even counts as being alive".

It doesn't, because the consent is given when the person is still alive and only applies once they are considered dead.

Ethically, it's by far the safest approach.

It doesn't solve any of the questions around the definition of death, only concerns about the treatment of dead bodies. The same effect could be achieved with an opt out system, instead of the current opt in one.