"Why should nearby property owners get a disproportionate say?"
Obviously, because risk and property damage potential are roughly inversely proportional to radial distance from the nuclear plant.
How about we give them a choice - ..."
In a civil society, there really is no choice about giving property owners a choice. For protecting life, liberty, and property is the only legitimate purpose of governance, no matter how that is implemented. Property owners out to some cutoff radius should be allowed to submit a vote (weighted inversely with radius) to permit or veto the construction of a nuclear plant.
This is the true point of democracy--geographically contiguous groups of property owners should be able to democratically administer the rights to use property in certain ways within their region. Democracy is illegitimate if it is voting on how much to take from a minority group at gunpoint, to give to a majority group, while a criminal gang gets to take a cut off the top. That is the "democracy" we practice now.
Coal plants are actually in a different category. Since they continuously pollute, they should have to pay royalties to the collective owners of the atmosphere. That's basically everyone on the planet, though there is a case to be made that due to circulation patterns, the payment should be weighted according to the statistical distribution of pollutants.
People should be issued a share of the atmosphere at birth. They may be traded freely once one reaches adulthood.
ALL pollutant emitters should have to pay, both individual and large scale. So the power plant will pay, that cost will be included in my bill. It will make the power expensive. Thus, a true market price (with externalities accounted for) will exist. Renewables and nuclear can compete on this basis.
However, the waste disposal for nuclear MUST also be accounted for!
When I buy gas, interestingly, the oil co. should owe no royalties. Since I will be the one doing the burning. So I will pay, reflected in the purchase price. But the royalties will partly get paid back to myself. This is fine. It also results in a true market price for the procurement and effective disposal of the pollutants resulting from burning the fuel. This will make it more expensive. But the royalties are not taxes, so they DON'T go to politicians who will squander them. They will go right back to the consumers.
This is fascinating because, some of the royalty cost cancels itself, but the increase in the effective market price of the fuel remains valid nonetheless! Government can't accomplish this. But they have a place in administering it--only to the extent of formalizing the definition and judicial administration of the property rights.