If you are employed you get a fixed salary to pay your bills. If you are self employed you rely on getting income for the work you book. She will have allocated and budgeted in accordance with the bookings she has. Late cancellations may give her time back, but not the money that she needs to live.
Yes, if she cancels due to her own illness, she may refund or credit. This is a cost self employed people have to absorb (and should be priced into their usual rates, along with holiday pay etc).
You are expecting her to reduce her income as a result of your own childs illness. That isn't fair and she wouldn't have a successful business if she regularly offered this.
Its the same with many hairdressers, beauty therapists, cleaners and other service professionals. The cost of living is expensive these days, and absorbing late notice client cancellations for any reason is not something any self employed person should have to do.
She has been particularly accommodating by agreeing to rearrange, so you should be very thankful for this.
To note, even if she has a regular employed job as well as the tutor work, it doesn't mean her employed income is 'nice to have'. It may be relied on to pay bills, be adding to savings, planning for future, debt repayments. Try not to presume or judge about her financial situation. If you enter into an agreement with a service professional, it is up to you to pay it, and your frustration to take if you need to cancel at late notice.