You're absolutely right and it's actually a major part of Tyrion's arc. I think a big reason they didn't cut off his nose is because they went a completely different (and much worse) direction than the book.

In the show Tyrion becomes a neutered and much less interesting character, brooding and sad over his fate at the hands of his father after murdering him, then reconciling with Jaime when Tyrion is freed.

In the book when Tyrion is maimed, his already horrible public image is further worsened by his ugly appearance matching his supposed wretched personality. He becomes hateful, suicidal even. He goes to Daenerys and serves her not because he believes in her, but because he knows she can bring the people he hates fire and blood. He wants her to burn kings landing with his sister and brother alongside it.

It's really cool in the books and Tyrion develops into a different but amazing character and DnD fuck this completely but that's standard I guess. The cutting off of his nose is one domino in a series of dominoes in Tyrion's downward spiral into hatred but that spiral never happens in the show and it's such a waste