Overuse of antibacterial products lead to bacteria that's resistant to it. So, the use of antibacterial products isn't the end-all solution. I would think that it would be better to stress hand-washing, sneezing and coughing into your elbow, and not sharing items. When a bug is particularly bad, I don't think using antibacterial items is a bad idea. But for general use, I'd avoid it. Save it for when it's really necessary.
There's evidence that children who grow up in an overly-clean environment are sicker later on. Their immune systems don't develop properly. They end up getting more allergies, asthma, and less immunity to the types of diseases one generally is exposed to in childhood. A little dirt and illness when young actually makes them healthier when they get older. Not that you want things filthy, but you really don't want it completely sterile either, odd though that sounds.
So, I hear what you're saying about the kids getting sick a lot, but I'm not sure that attempting to create a completely sterile environment with antibacterial products is entirely the answer either, except, of course, if the illnesses are severe or rampant.
Also, there's the matter of cost. If they only have one cleaning person and are using only water or vinegar, it may be a funding issue. They may not be able to hire more cleaning staff or buy more expensive anti-bacterial cleaning products.
Since it's a private school, vote with your pocketbook. Get some parents together to demand a change or you will leave. One or two students leaving won't affect the school much, but if a bunch do it, it will get noticed. Make sure you understand that tuition may need to go up to pay for it, though.
If a tuition increase isn't acceptable, perhaps, as a community service project, a group of students and parents could volunteer to disinfect classrooms. Or purchase better cleaning products. Or raise money to hire another janitor.
Depending on where you live, public schools can be wonderful or horrible. If you're in a decent district, public school is just as good as the private alternative and it's free. Of course, if you're not in a decent district...