This, I am not gonna argue with any stylist about MY hair. If they disrespect, I am out of there. They are there to provide a service for pay. Your opinion is the only one that counts.
I personally don’t care because they can’t ruin it and it always grows back. I rarely see the same person.
From my understanding, depending on how the stylist was employed by their shop, they might not be able to contact their regular customers when they leave. You kind of have to track them down on your end. A bit of a rude surprise but it can be a cutthroat industry at the high end.
Appointment? Stylist? 
The last time that happened I just asked where she had gone and then went there. If she had retired or left the state or for whatever reason was no longer available to me I;d give the new girl a chance.
OTOH, my hair isn’t particularly complicated. Kind of a longer military buzz, so not real easy to screw up.
Another really don’t care at all about who cuts my hair. I have no interest in a fussy hairstyle that takes effort to maintain. Just keep it out of my eyes and long enough that I can put it into a ponytail during exercise, thanks.
I think there’s another option - let the person cut your hair that day but then try to track down your old stylist if you really liked him/her. I did have something similar happen to me but my regular hairstylist got in touch with me before hand. She is absolutely wonderful- she does a great job and I like her chill, gentle manner. She had a very serious illness and was out of work for about 4-5 month and she set me up with the other gal that works on the weekends.
She was awful- I hated her personality (Trump fan) and instead of doing highlights, she slapped dye on my head so it burned and basically did the same color. She got my appointments wrong and accused me of not showing or seemed put out that she didn’t schedule enough time for what I usually had done. She almost ripped my hair out of my head when she brushed it. Just an unpleasant, sloppy person. In retrospect, I should have gone elsewhere but I’m very awkward with the whole hairdresser thing and I really wanted to be there when my hairstylist came back. So if my hairstylist disappeared again, I’d try to track her down and then I’d go somewhere else entirely so I wouldn’t have to deal with Gorgon hairdresser again.
I’ve only had a few stylists in my life who can adequately deal with my thick, wavy hair. If I made an appointment with my usual stylist and I wasn’t notified in advance that she wasn’t there anymore when I showed up for my appointment, I’d be pissed. I’d walk out and find out where she went and make an appointment with her. I’ll have to talk to her at my next appointment and see if she’s allowed to tell me in advance if she should happen to leave. I suppose some salons have contracts about that kind of thing.