I agree that it depends on the individual. I started a pet sitting business at 10 and only received one call from the numerous posters I put up. Looking back, I can't believe I even had one client. In general, young people are not responsible, have little motivation, and just want to get the job done quickly. Young people that work are usually in it only for the money and have very little care about the quality of work.
Grooming can be a scary place for some pets and people don't want to take their dog to a groomer that is not a professional. Most professional groomers know how to handle dogs in order to get the job done well and without fear. For people that have their dogs groomed on a regular basis, it is important that they have a good experience so that the dog isn't afraid. Additionally, many breeds have "standard cuts". You will need to be able to trim the dog to the owner's specifications. If you cannot do that, they are not going to come back or recommend you to friends. They may not even pay you if they are unhappy with what you have done to their beloved pet.
If you can somehow prove that you are trustworthy and know how to handle someone's dog, you may have a chance. Personally, I think it is a bad idea. Why don't you try dog walking and then ask your clients if you can bathe and brush their dog?