The main reason, as far as I can see, is that the dividends are payments with which the shareholders may do what they want.
Capital that the company has no use for does not make a significant positive return on investment, as you pointed out, yes the company could accrue interest, but that is not going to make the company large sums of cash.
While yourthe company may be great at making shoes - maybe even the best in the world - doesn't mean they are good investors. Sure they could dabble at using their capital to invest in other equities, but they don't, because they just want to focus on making shoes.
If the dividend goes to the investors, they can do what they wish, be it reinvest in the company, or invest elsewhere. Other companies that may make good use of the capital, and create significant returns on it are one such example.
That is the rational answer, beyond that, one of the main reasons is that people like the feeling of receiving dividends - it might not be the answer you are looking for, but many people prefer companies that pay dividends for no rational reason over companies which grow their asset value.