Your question did hit home, thank you.
The answer is really in two parts. Firstly I can and have seen it as a weapon, in training and outside of training complete with bruises and accidental damage caused. So contrary to what some may think I have no problem with using it as such, watching others use it as such whether in another martial art or the various Aikido styles.
You could say I used it recently as a weapon when doing some security in a pub. Actually it was a pool cue about to be used like a club to batter someone being held by two idiots. I grabbed the end and pulled it down to waist level spinning the guy round to face me. I was in a good place at the time and so came out with a bit of a wise crack saying to the guy not to use it until he knows how. He gave a few expletives and pulled the other end of the cue to try and take it from me. It was too perfect. I moved foreward with his own force, extended through the cue and did a quick hard nikkyo with it's end and he hit the floor hard.
His friends were too busy laughing at him for a moment which allowed me to then use that opening to diffuse the situation.
To me I thrust through like a spear and then cut like a sword, thats just how see it.
So that doesn't mean now I see it just as a weapon as I see it as a tool of Aiki.
In fact I see it as a friend, a dancing partner.
It sounds as though you feel it is hazardous? I don't intend any disrespect with this comment. Your anecdote is one where uke has the jo. You describe teaching students to be with the jo. You describe it as a partner. I'm simply saying you don't seem to regard it as an extension of yourself in the way that a weapon must be.
I'm very clumsy with the jo myself, but I've made a lot of progress with the sword lately. I can tell you that kata training, assiduously, under the tutelage of a good instructor DOES work wonders with regards to weapons. As your level of comfort and proficiency increase, it opens new doors for expression of your Aikido.