I agree, stick to your guns. I often feel really bad about charging for lessons, but I treat it the same as private education. If you take a day off school or a day off uni because you're ill you can't expect a refund.
I have a very difficult situation at the moment which I would appreciate people's advice on:
One of my students has anorexia which her mum has told me about. Her mum has cancelled 2 out of the last 4 lessons and about 10 mins before the lesson. Would people charge or not??
I also have a waiting list and am wondering if she should take a break if it is too much and allow someone else to take the slot. Very complicated.
I would explain that missed lessons must be paid for, and you feel bad for them. So perhaps they can move to an ad hoc slot. Although I suspect if it is mental illness, they won't get any notice. It'll only be when mum says it's time to go that she melts down.
I would still charge though, unless they are last in the day, you can afford to be flexible, and you can do something else easily if they don't come.
For me, I have to arrange childcare and pay for it, so all my students pay whether they give notice or not.