Most consultants have pagers or cell phones that will allow you to reach them wherever they are. With the consultant on site I get to see every hour they're charging. The one you have has failed to earn your trust, discussing each step of the process, providing ongoing estimates and reporting of the progress of the engagement and the time it is taking. No, more likely, it will be just the opposite. Your consultant's daily presence will not of itself, speed up your gathering of information, or management's decision making process.
Successful consultants have existing clients with ongoing needs.
The consultant can juggle these demands and priorities at their office, or try to handle them at your's, or put them off.