Why is it bad to download the full chain from a third party with Bitcoin Core? I know Bitcoin Core downloads and processes the full chain much quicker now than it used to but what exactly is so bad about downloading a compressed blocks, chainstate and database folder from a third party?
In the event of any malicious changes to the chainstate, wouldn't Bitcoin Core at its startup reject the new blocks anyway?
No, the chainstate is the nodes view of correctness. It would take many hours or days to re-verify the whole chain on most hardware, so an assumption is made that the state stored at the previous shutdown is most likely correct. There's some sanity checking of the previous couple of blocks to make sure there is no gross accidental corruption, but malicious modifications of the chainstate before that time will not be caught.
A maliciously modified chainstate can contain fake outputs that do not exist in the canonical one, meaning you could be duped into accepting a false transaction paying money to you that nobody else would consider valid. If any significant portion of the network was downloading unverified chainstate databases it would be a considerable issue.