I used Lucas Red n Tacky, but I am not satisfied with it because after a couple of years it starts that running mess too. I am sure others will chime in with their experiences.
I've had the same experience with Lucas Red n Tacky. That being said, I still think it works better than most.
Disc brake grease will do well not running or melting, but I don't think it's best for the application. Not enough heat generated to meet its qualities in a neck bearing
you're right its 10k x overkill. The only purpose of greasing the neck is to keep dirt out and from running dry.