What would be an advantage of buying into a franchise when, in all aspects, you have the financial stability to just start your own business?

The main reason that people buy into a franchise system is risk reduction. A strong franchise with a proven track record of success that you can verify through your own investigation is a much safer path to starting a successful business than doing it all yourself.

A good franchise is going to be able to tell you the most effective ways to set the business up, the best ways to market and staff the business, and the best practices to run it on an ongoing basis. Though you may be able to do many of these things on your own, you'll probably go through some fairly expensive trial and error before you get it all figured out.

There's a lot more to starting a business successfully than simply having enough capital available--though this is certainly very important.  You should carefully think through all the other aspects of what you need to do (writing a detailed business plan is always a good method to do this) and how you are going to do it, then decide whether an established franchise system is a good investment for you.