The Spock/McCoy thing worked because McCoy was the provocateur in the relationship - he was, at least in principle, the less authoritative of the two characters and he was challenging the chain of command. Spock would presumably have been satisfied to treat the Doctor as he did the other lower-ranking officers with whom he dealt - Uhura, Sulu, Scott etc - impersonally and with respect.
Gender politics aside, Harris is the more powerful character here. Having her kick Cutty every few minutes makes her look both abusive and insecure. I'm embarrassed that I didn't understand that when I was writing the thing.