If a salaried member of management has used their personal/sick time and then misses another day of work, can the employer not pay them for that day?

Yes.

Absolutely. If they have no more personal time or sick time available I would expect a company to deduct that one day's pay from the employee.

Of course. That's why there are personal/vacation/sick day limits - or an employee would take off as much time as she/he wanted and get paid for it.

This particular employee states that because they often exceed 40 hours a week for travel purposes, the extra day should be excused... they are an "exempt" status employee.

This particular employee states that because they often exceed 40 hours a week for travel purposes, the extra day should be excused....they are an "exempt" status employee.

I think you have to post the entire question and circumstances in one place in order to get a correct answer. Otherwise - it's advice, more facts, more advice. Makes no sense.