It wouldn't be fair to generalize any statement starting with "all personal chef" this or that. That said, there are a few qualities that would help a personal chef excel in his or her business over others without these qualities.
First, it sure does help to be personable. You've probably met or even have friends who have said "I don't like people". Truly, that just means they don't have good communication skills and have difficulty expressing themselves or understanding the expressions of others.
Enjoyment in Helping
If you don't enjoy helping others or have patience when it comes to working with people, you may not do well as a personal chef. Our profession is all about meeting people in their homes and making them feel comfortable with the fact that the chef is going to likely be all alone in their house, often having access to door codes and other private information. Making others feel at ease is often something that comes naturally to those who enjoy helping others.
Attention to Detail
Having attention to detail is not essential to being a personal chef, but goes a long way in the realm of customer service. If you remember that the Jones' love olives in their spaghetti sauce, it's going to make them feel special. It could be that you thought to put those notes into your recipes, not that you actually "remember", but this little attention to detail will mean a lot to the family you're cooking for. Little things like putting the tea kettle back on the stove just how it was when you arrived shows that you know what the family prefers and can accommodate.
Being adaptable is probably one of the most important qualities of being a personal chef. I really don't think you can run a successful business without this. You may show up and only one out of the six burners are working. You must adapt for the day. One day, your client may announce that they're vegetarian. Well, now it's time to adapt a new menu for this client. You see what I mean? If you throw an internal tantrum each time things aren't perfect, you most definitely won't last long.
Are these all qualities you possess? Or is it that you think these qualities are not important as a personal chef?