You must in fact have your own gear. If your gear is not in good working order then it's not pro gear. If you need to borrow a string or a battery from a potential new band mate to proceed with an audition then your gear is not in good working order (see above). I've seen this happen more than once. It's not a fun time.
Great point. Sufficient gear for the job includes extra strings, batteries, picks (if needed), straps, cables, and tuners. Years ago I played in bands with younger guys, who were generally great dudes and fantastic musicians, but I got tired of supplying batteries, straps, cables, picks, etc. because they "forgot." I always did though, because I wanted the gigs to go well.
In the second band like this, the guitarists didn't have great instruments, so they went out of tune frequently. And yet they didn't have pedal tuners. (This was before clip-ons.) It was brutal. Before our first gig, the keyboardist and I ordered them to buy tuners, and said otherwise we wouldn't play the show. They bought tuners, and all our gigs went relatively smoothly from there.
When classifieds say "pro gear," I think they're referring to instruments and amps, but for gigs it goes beyond that IMO.