Well, I have one child instead of three, so I have it easier in some ways (less sibling bickering) and it can be challenging to be the only adult around most days.
Yes, I would have put the kibosh on the fun if the kids are acting up. Heck, we had to do that on Saturday; got to where we wanted to go, some unpleasant not-listening started, we warned, received uncooperative behavior and my husband and I decided that going home was the better option. And then, he spent an hour in his room, which was a very real consequence: if you are going to be so unpleasant that we can't enjoy ourselves during an outing, then you may not be with us.
The rest of the afternoon/evening was far better. I am a tough mom sometimes and extremely consistent.
Your twelve year old can get hired on as a mother's helper or a babysitter. The Red Cross usually offers classes for this. Let her earn some money for herself. Have all of the kids start a 'summer reading program' if your library runs one-- have them keep track of the book they have read or the amount of time spent reading/being read too. Common goal for all the kids.
When you do your 'jellybean jar', don't make it so hard for everyone. Set the goal and then, when the jar is filled, go. Don't make it contingent upon the jar being filled by a certain day-- that's too much pressure. I'd also let the kids 'award' jelly beans to each other by noticing the nice things they are doing for each other. This helps them to LOOK for the nicer behaviors instead of focusing on the negative. We do something similar at home where Kiddo has to earn 100 points and then we do a family activity/outing. Sometimes he earns them quickly-- at other times, we are all challenged and need to refocus on the small things he is doing *right* so he's not getting inundated with negatives. And he knows that if it takes two months to earn it, he'll still get to earn it, and he'll still keep working at it.
Like Doris Day, if we complained about being bored or got sassy, we were given chores. Like Laurie, at 12 I was the oldest and had responsibilities around the house and in keeping an eye on my younger brother (and NOT arguing with my sister, which was tough but my job). We were not entertained, even at seven or so, I remember my sister and I were pretty much left to our own devices and mom made meals. Sometimes she played a board game with us, but there was no nonsense and we were expected to get along.
I like the idea of schedules: when I was a nanny and had groups of kids for the whole day, I had a daily routine, times for the park (morning, when it was cooler) times to do art in the basement (hot afternoons), times for rest/reading, times for coming together to do a project or play a game. I didn't have tv or video games to fall back on as a nanny, so those days were structured enough to have a good balance of independently-led activities as well as the meals, snacks, and other reasons to come together.
My sister has three kids and she separates them all for one hour a day in the summertime. She has projects for them if they are bored.