^I think your quote makes my case for me. Spock's first
recommendation is to strand Mitchell on Delta Vega. When Kirk rejects it, Spock says that killing him is the only other choice. And his argument convinces Kirk to overcome his resistance to the recommendation to strand Mitchell.
And bringing a weapon as a precaution does not even remotely prove an intention or preference to use it. Cops carry guns on every call, but most cops go through their entire careers without ever actually having to use their guns outside the firing range.