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After some discussion with Tony Djukic, I attempted to restore my backup with a different plugin, Duplicator. This particular plugin creates its own fresh install of WP. Having run that script, everything in the site is working fine as of me writing this. It appears that using the old version of All-in-One WP Migration caused some sort of problem, but I can only speculate what exactly. It seems to have failed to properly update the DB, but I am not sure. Regardless, I found a solution that works, even if I don’t know why, so for now I’m happy.