That seems to be basically the same issue as reported for mutagen (the tag reading component Picard uses) at https://github.com/quodlibet/mutagen/issues/496
That issue is about WAVE files, but AIFF and WAVE files share the same basic structure and they share some code in mutagen.
In my (phw on Github) comments there I mention that probably mutagen should just fail more cleanly. That would be the easiest solution, but would not change much for your situation with Picard. The files are in a way broken and loading would fail.
But maybe we could detect cases like this, where the internal structure and the sizes noted there are fine but the file is missing some bytes at the end, simply assume the missing bytes as null bytes and add them. I haven’t looked into this in detail yet, though.