It is strange that there are still business owners who would rather travel with their package and the courier in order to ensure that it safely reaches its destination. This is not a service that is typically offered by any local courier service, and there are a number of reasons why this kind of service has not been made available to people. The basic line of thinking as well, is that if you don't trust the courier to effectively take your package to your destination then you should either move it with your own staff, or you should hire a private driver to take you and your package to its destination. A courier company is not a taxi cab company, nor should it be confused with one. The job of a courier or delivery company is to deliver goods and packages only, and it is not in the driver's job description to have to accommodate a passenger en route to a delivery.
In addition, there are legal considerations that would prevent almost any reputable courier service from allowing a client to ride along to a delivery destination with the courier. Courier companies are required to carry very high levels of insurance in order to stay in business. They must carry insurance on all their vehicles, as well as courier insurance that will cover all of the goods that they are transporting. By allowing people who are not employees of the company, and who are not bonded, into the vehicle, they could be voicing one or many points of their insurance which could leave them open to tremendous amounts of liability. Unless the courier company that you deal with offers this as special service that they advertise, it is quite likely that any delivery driver would lose his job for allowing a client to accompany him to his delivery destination on a normal delivery route.
When making special requests of a local courier service, most companies are going to do their best to meet their clients' needs and requests. However, a courier company is not going to risk compromising their insurance, which is potentially protecting their entire business, to accommodate the request of a single customer. If you are not comfortable letting your courier take your package to its destination then you would be best advised to look into alternate means for having it transported that you can be more satisfied with, or that will allow you to travel with your delivery.