It is worth saying that the lyrics of many successful songs are either cliched, have minimal meaning, or don't mean anything at all.
That isn't necessarily a criticism. Often lyrics are intended to create an atmosphere or convey an emotion and don't tell a story of any kind.
Clearly, lyrics are important in the songwriting process, production and recording. Firstly, the lyrics that are sung must be correct, particularly if the song is a cover. Also, the lyrics must be adequately intelligible, in the context of the requirements of the production.
But are lyrics important in mixing? I'd say yes, and for two reasons...
Mixing should not be considered a merely technical process of balancing the tracks and coming up with a sound texture that would satisfy the needs of the market. Mixing is an artistic process, that a good mix engineer will support with a technical proficiency that is so fluent that he or she doesn't have to consciously think about technical matters.
So to create a mix that truly supports the song, then it is vital that the mix engineer understands what the song is about, communicating with the writer ideally on any matters of doubt.
Obviously the mix engineer can hear the lyrics in the vocal track, but having a printed copy to hand is bound to help. And I'd say that a printed copy is better than electronic, both for ease of viewing and for scribbling any notes or comments.
I will add at this point that there could easily be a mistake in the vocal that hasn't been caught during production and recording, and the mix engineer is the last line of defence to throw the issue back at the writing and production team.
The other reason it's important to have a copy of the lyrics is that the mix engineer has to judge how intelligible the lyrics need to be. For some songs, the listener needs to understand every word without effort. For others, the lyrics are more like a cloud of words that prompt the listener's emotions but don't need to be heard in detail. A printed copy of the lyrics will most certainly assist the mix engineer's decisions.
Of course, there will be some mix engineers who prefer to work without a copy of the lyrics and if that's their preference then that's fine. But I would say that for most mixes, paying attention to the lyrics and having a printed copy can only be a good thing.
Great home recording starts with a great home recording studio. It doesn't need to be expensive if you know how to select the right equipment for your needs.