The second half of the paper, "Linking with temporal analysis". If you read the second half of the introduction, you will find that the primary technique they use for tracing 80% of transactions is found in the current version.

The sloppiness of this code is really shocking, "when the Monero client chooses mixins, it does not take into account whether the potential mixins have already been spent."

That's because RingCT removed the ability to create a ring signature with those outputs, so adding a complex whitelist / blacklist mechanism would have been a massive waste of time.

This is blatantly false, and I implore you to do further research before making and spreading such conclusions.

I'm not thoroughly familiar with monero's internals, so someone please correct me if I'm wrong, but I thought it was well known that this was a deliberate design decision. Previously spent amounts don't actually run a risk of being double spent as they're only used anonymization purposes, as far as I understand. So why is this is considered "sloppy"?

The results of the mitigation are shown in the paper as Figure 5. The success of the techniques in the paper decline rapidly over the course of 2016 and would effectively reach zero if the dataset were extended (this is noted in the text when it states that RingCT transactions are immune, although even without RingCT it would still effectively reach zero)