I know that some folks in paleoclimatology are interested in these core methodology problems. But these concerns are not conveyed to a wider audience. When a temperature reconstruction graph is released to the public there is no mention at all of these problems and one can easily get the impression that these reconstructed temperatures can be known almost as accurately as going outside and taking a reading from a thermometer.

Can the public understand that?

Excluding math, we use proxies for just about all science. With different assumptions and different criteria. I don't deny that there are politics, assumptions and margins of error but they do put effort in to minimizing them and doing real science. Take a look at pharmaceuticals, they don't even know why a lot of medicine works, in fact there is debate on if it even does... But people still buy it and take it.

The public can't understand that, but it certainly forms opinions based on it. It votes, too...

The public is, honestly, incapable of understanding, or even caring, about underlying issues in proxy temperature modeling. A large amount of the public is incapable of distinguishing between weather and climate.

Proxy temperatures, statistical methods, atmospheric modeling, etc, are all scientific areas whose knowledge can be gleamed at sufficiently well for a person with some form of higher education in science or math education (but only barely; remember that this stuff is being researched by people who have devoted their entire lives to the scientific specialty). But that is an astoundingly low proportion of the population. And even then it is clear that, as methodological errors have diminished and methods for more accurate models have appeared, that they all tend towards the same conclusion.

We should be worrying significantly more about the small but non-trivial (and ever-increasing) probability of an extinction event within the next decades rather than pretend that methodological errors, at this point, are a significant issue in identifying the climate trends we have spotted in the last decades.