If I have to pay you not to dump toxic waste behind my house (but not on my property), it makes no sense at all to say that voluntary action has "solved" the externality. Maybe the difference between the external cost I would have experienced and the marginal value of dumping the waste can be said to have been "solved" - creating a market where there was none internalizes that portion of the cost.
But we don't like externalities because I'm still bearing the cost of the marginal value of the dumping the toxic waste in the first place. In a sense you could say it is internalized into an unjust/perverse/extortionary market relation which isn't really "voluntary" at all. Why? Because it just shouldn't have been dumped in the first place.
So it seems to me arguments like this (and maybe there are other arguments, but of course this is one that you commonly here) aren't solutions to the problem of externalities at all.
If I can't say the bolded sentence above. If we are instead dealing with an example where I can't legitimately claim that I have moral standing and the polluters don't (and certainly there are cases where standing is shared), that doesn't solve anything either. The person taking the action can still extort that portion of the benefit. If the land that the toxic waste is being dumped on is land that isn't owned by anyone, the polluter can dump and demand payment to stop but I can't require payment before dumping occurs.
Standing is subjective - they may feel they have the same standing that I do, perhaps. And they can make that case.
But you can't make an objective claim that the externality is solved by the market in this case if I don't share that understanding of our respective standing in the transaction.
Property rights are nice because they help these transactions to function, but property rights don't provide a self-justification. We're perfectly entitled to reject the legitimacy of a property rights arrangement and no one can objectively challenge that (again, they can of course subjectively challenge that rejection).
The last three paragraphs ought to get you thinking about how this relates to the problem of social welfare functions and interpersonal utility comparisons. To get anything meaningful out of those exercises we need to make assumptions about weighting utility. That's fine to make those assumptions, you just have to acknowledge it's a normative exercise and not a positive one. The same goes with these claims that markets (in this set up) solve externalities - you can say that but don't pretend you're making a positive claim.
Where am I wrong? This is just a set of thoughts I threw together so I may be.