The flaws are numerous, but the easiest to explain is that they employ a post-hoc screening bias. Imagine that you gather a few dozen time-series together. They go back two thousand years. Then you regress them against the (very short) instrumental temperature record in the period where they overlap. You keep the ones that pass and throw the rest away. You form some sort of linear weighted average of the selected proxies (i.e. PCA for example).

The thing about this process is, if you feed in completely random red-noise time series you will get a hockey stick. The reason is simple, the selection period will mirror the instrumental temperature (mental check: it has to, after all we only selected the proxies that matched the instrumental temperature). You have your blade. But none of those red noise proxies actually reflect past temperature do they? They just happened to be selected by random chance. So in the time before the selection period, the noise cancels out and you have a flat line. There is your stick.

This is the core of how these "proxy reconstructions" work, and it is nonsense.

Outside of that, it gets even better. The stories are comical and numerous. Mann famously fed a contaminated Finnish lake sediment series into his algo, which happily regressed it against temperature and found that it passed significance if it was used upside down (in the opposite orientation to that suggested by the author in the original source paper). So in it went. GIGO. Or how about the question of which temperature series to regress against? You'd think you'd regress against local temperature right? Or maybe at least the same continent? Oh ye optimistic fool, go and google "teleconnection" and "non-local statistical relationship". This is what passes as "climate science".