I own and operate a preschool and daycare center. I charge by the hour because most of my families work by the hour. The state pays for the title XX kids similar to what you do. Anything between 6-9 hrs. is a flat daily rate. Anything over or under that time is billed per hour. It states in their paperwork that they only pay for time the parent is at work and the time it takes to drop the child off and get their to pick them up after work. If parents are doing something else than work, the provider legally cannot charge them for that time and they are charged privately. (Actually getting the money from some of them is another story. )
In my experience, anytime a people are put in a position of a flat rate, it allows for someone to take advantage of it. For example, many people, once put on salary at a job are expected to work longer hours without additional compensation. Much the same way, by having a flat fee you are putting yourself on salary and the parents are milking it for all they can get. It's kind of like, if you have to pay so much for something anyway, you might as well make it worth the money.
So, anyway, it maybe more work but may contract each family based on their work schedule and any time over the hours contracted will be billed at so much per hour.
Personally I don't have a big problem with parents running errands and doing things like mowing the lawn while the kids are in my care, as long as it's on occasion. Mainly because I've been there, done that, trying to accomplish tasks and having kids underfoot. Mowing the lawn, actually is kind of dangerous to be doing with kids if there isn't an adult to supervise them. What bothers me is when parents know I close at 6, they are done with work earlier than that and leave their kids till 6 just because they can. It almost never fails that when they do that my 6 oclock families are gone and I could have been done most of an hour earlier. It also bugs me when it's stormy or a holiday and parents don't stop to consider that I may want to get home early. Those are usually the times that parents are late!
Hope some of the other gals will have some helpful tips for you. Good luck!
Wouldn't it be nice if every time we messed up our life we could press CTRL ALT DEL and start all over??