What would you do in this situation? A candidate comes into an interview chomping on gum like a cow chewing grass. Do you ignore it and conduct the interview like normal, ask the candidate to spit out the gum even though it makes you feel like a middle school teacher scolding a student, or do you conduct the interview even though you've pretty much already written the candidate off because they clearly lack basic manners?

Just something to consider, chewing gum can be a method used for multiple disabilities. An Occupational therapist will utilize things like chewing gum to help people with tiks and even ruminations from OCD. I can personal speak on a family member that would have snacks that were as "crunchy as possible" throughout the school day to help with tics. Yes, it was annoying but necessary. During an interview? Maybe, obviously that would be a time where they are the most nervous. However, I feel like it would behoove them to say right off: just to let you know, I apologize for chewing gum during this interview, it's necessary for a medical condition. If it becomes too distracting, just let me know and I'll do my best.
Because Obvioulsy, as we can see here, people will judge their job qualifications if they are chewing the gum.
Myself, I don't ever chew gum unless my TMJ is bad from a night of teeth grinding. PT suggested chewing gum to move different muscles and it helps tremendously. Actually relieve pain without medication.
Just something to consider.

I would take it as a way to see how they handle negative feedback. "I'm finding the gum chewing to be distracting. Would you mind throwing it out before we continue?" and see how they behave from there. Though, it would be hard to come from. They should have known better.

Do the interview and assess the candidate on the competencies. That’s what you’re there to do.

Oh, by the way, I did the third option and I don't feel even a little bad about it. I wrapped the interview early, too. It was so rude. I can't believe how clueless some people are.

I wouldn’t have asked them to take the gum out either. But I probably would’ve wrapped up the interview pretty quick. There are some basic decencies people should be observing in an interview.

I would either ask them to spit it out so we can understand what they are saying, explaining my why, or ignore and be pleasant. I think as recruiters we forget they have the power to also review your company on job boards. So as good stewards of the company you don’t want to ruin a companies reputation by also being overly critical and rude. They don’t need to know you find this rude or unprofessional. It’s not your job to teach them.. I say this more for your own personal mental health, not that you did anything wrong.

I would continue the interview but note it. If everything else is strong, it may just be nerves. If multiple professionalism issues appear, the gum becomes part of a wider pattern.

I’m with you on this, but I would finish the interview & send a rejection message. Once, I was recruiting for a janitor position and the candidate that came in with a graphic t-shirt, ripped jeans, and black slippers with fur. Then when she spoke, she kept clicking her tongue. We were a panel of 2. My partner kept asking question, I stopped and was simply doodling on my paper. She was not selected for obvious reasons, bad manners and not presentable. For me, it doesn’t matter how low or high a job is, be presentable. I understand if the candidate came in their uniform of current employer when they just finished their shift to make it to their interview (this happened, too.)