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.
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.