The point of this warning is that you cannot trust any information before you've validated it. Note that the underlying mode of AEAD ciphers is usually CTR mode (other modes are similarly affected). So an attacker can for instance introduce errors in the ciphertext which translate into errors in the plaintext at the same location. Most AEAD ciphers use CTR mode underneath, so the attacker can flip specific bits of the plaintext that way.
An attacker can for instance learn about the plaintext by watching for the specific errors to occur when the - now invalid - data is processed. This is what the warning is about: you first need to establish the integrity and authenticity of the decrypted data before processing it.
Of course, this usually means caching the data until it is validated. For that reason it makes more sense to first create a buffer and load it with the ciphertext. If you have a good API you could overwrite it with the plaintext so you don't have to allocate the storage space twice.
You could still store the plaintext on disk of course, as long as you make sure to remove access to the data if the tag is not valid. For instance, you could store the data in a temporary file and then rename it to the right file name after verifying the tag.
That all said, no, the AEAD mode is more secure than any other, non-authenticated mode of operation, as you can verify message integrity and authenticity. But you can still use the cipher incorrectly, and that's what this is all about.