I thought the reason was that a virus has no means to reproduce without hijacking some other living thing's facilities with which to replicate itself.
We want to exclude viruses because they are too mechanical. There's no magic to them. That's the way it appeared before we started finding all these giant viruses anyways.
But the presence of a cell wall is a pretty significant feature nonetheless.
It's similar to how it's difficult to argue whether a data format is a programming language (Turing completeness is an obvious but unnecessary and insufficient distinction). "Declarative programming languages" muddy the waters but most people would agree that a JPEG image is just data whereas a PHP script is a program.