How to respond to an upset student?

I would not intervene unless I felt the student was at risk of harming themselves or others.

As you say, college is about more than grades, learning how to handle failure is part of life.

Moreover it’s not unlikely that their performance on the final was the icing on the cake: they had a bad day/week and were upset about all of it plus the final.

Unless you’re willing to allow them to retake the exam (which is probably a bad idea unless they offer a good reason), reaching out to them and telling them they should deal better with exams will likely annoy them and demotivate them further.

It may well be that the student was upset due to frustration with their own performance and had nothing to do with you or with being cut off. Their actions may not reflect unprofessional behavior as much as just immaturity. Frustration often boils over into anger and the anger is often misplaced. You may have experienced this yourself, actually.

I don't think you need to be proactive here, though it is fine if you want to be. But your best response is to just listen, even if it is to a rant. If you think that some encouragement is needed you could give that, as well as offering to give suggestions on how to do better in such courses.

But you can leave the next move to them, I think.

It is normal for exams to have time limits.

It is also normal for exams to be stressful for students, and thus I would not worry much about a minor lack of professionalism in this context.