Is the word "broach" old-fashioned? Or still in use today?

Its not in common usage here, in western Canada. You'd more likely hear approach the subject. It may be old fashioned, I don't know, it may be very common in other places.
Many people go through multiple interviews for a job. The first interview might be with a human resources person. It would cover the basics of the job and try and fill in blank areas on the resume, such as lengthy periods of unemployment or specifics eg. computer languages, or if you are willing to relocate or work nights. This interview is quite frequently done over the phone.
A second interview would be with a manager or a board of 2 or more people. Sensitive topics from your past (personal history) could mean many things from health or mobility issues, religious requests - not working on sundays, a criminal record and so on...
Once you pass this interview, you may get the job or you might have to go through another interview.

Thank you Ceili.

I would use the word - but I'm an american from California, and an older woman.

edit - I don't think I'm the only one who would use the word, even if it is going out of style.