The main problem is that no one knows if the hypothesized models (temperature vs. tree ring thickness, sediment layer thickness, etc.) have any validity. Why should tree rings grow linearly with temperature- what about a huge number of other variables? The temperature “signal” is at best very weak. Decoupling the temperature signal from other variables (humidity, elevation, tree species, CO2 concentration, etc.) is a hugely difficult multivariate statistics problem and most of the researchers I've corresponded with do not seem to appreciate this fact.
Another major problem is that proxies get weighted in the temperature reconstruction based on how well they fit the hypothesized temperature model. The weighting is where things get really shaky. Right away, you have biased your results in such a way that that nonconforming data is discarded / de-weighted. Note that there is not anything wrong with these nonconforming samples that invalidates them; they just don't fit the proposed models very well. Typically researchers develop their own methods for weighting samples in the temperature reconstructions and many of these methods would make a statistician cringe. Often during the weighting phase you find that a handful of proxy samples (out of a sample size of hundreds) get the vast majority of the weight in the temperature reconstructions. Sometimes the weight of a temperature reconstruction in a location is almost completely computed from a few proxy samples taken halfway around the world at a completely different elevation.
If you try to discuss these fundamental problems with climate scientists you'll typically get told that you're not a scientist (not true) or that you're a "denialist" being paid by big oil (not true). I am convinced that these temperature reconstructions are useless at best.