I researched, and this is quite hairy:
-a is deprecated, thus isn’t listed in the manpage for /usr/bin/test anymore, but still in the one for bash. Use -e . For single ‘[‘, the bash builtin behaves the same as the test bash builtin, which behaves the same as /usr/bin/[ and /usr/bin/test (the one is a symlink to the other). Note the effect of -a depends on its position: If it’s at the start, it means file exists. If it’s in the middle of two expressions, it means logical and.
[ ! -a /path ] && echo exists doesn’t work, as the bash manual points out that -a is considered a binary operator there, and so the above isn’t parsed as a negate -a .. but as a if '!' and '/path' is true (non-empty). Thus, your script always outputs "-a" (which actually tests for files), and "! -a" which actually is a binary and here.
For [[, -a isn’t used as a binary and anymore (&& is used there), so its unique purpose is to check for a file there (although being deprecated). So, negation actually does what you expect.