It's his job because he has made it his job. His very behavior revealed that explicitly. Fran doesn't give a damn about his job, nor does great many other adventurers, I bet. From Fran's pov, the dude is someone trying to hold her back and slow down her progress toward her dream. All she needs is the sword.
It's her business. She's the kind of person who rushes forward and subjects herself to deadly dangers to grow stronger. She would be dead without Teacher, but she has it, so that's it.
They can be worried about her and push forward, but they should have advanced rapidly anyway, Fran or no Fran. The guild is a collection of independent entrepreneurs, who decide for themselves what they are doing. Sure, it's humane to try to look after someone, but at the end of the day, they are all responsible for themselves and have got no one but themselves to blame if they meet an unhappy ending.
They knew she had already earlier wiped out an army of goblins, alone, saving that group of adventurers she sent back to give the report, while Fran herself remained to slaughter the goblins.
It was their decision to make haste, to chase after Fran. They could have as well progressed normally, at whatever snail's pace was their plan, and leave Fran to deal with Fran's problems, since it was Fran who decided to leave everyone behind. That would have been the more responsible choice, but since they didn't do that, I reckon they didn't actually risk much, contrary to your estimation.