However, I am curious about when the child would practice. If the weekly schedule has only a single hour open for lessons, then when is there space in the weekly schedule for practice?

I was thinking over, just yesterday, the small danger signs that I had let go by, early in certain relationships.

Turns out, I should have paid more attention; the signs were right there. And I say the same for this prospective student's parent. I have found the comments interesting and worthwhile, but since you say you see the potential for trouble yourself, I think you have also seen the answer for yourself.

Also, I would not ask the current student if they wanted 9 am or 11 am, because it still implies that you want them to take this other time, or you wouldn't offer.

I find all the conjecture and name-calling about this mother rather unfortunate.
We know absolutely nothing about her or her situation
... ... ...
The advice to simply put her on the waiting list is best, IMHO. She and her child could be wonderful. You just don't know right now.

This 9am student has been with me 3 years.

you can't expect the slot to open up within a couple weeks or months.

....so I would encourage you to see if you can work that into your schedule, and move into a more desired time as slots open up......

"I can wait as long as it takes. I only want Saturday 9am slot."

Meanwhile, a parent with this attitude could still be waiting until the sun explodes before I would accept her into my studio.

WHAT attitude? The thing that bothers me is the interpretation and extrapolation. If someone says A, it means all these other things. A customer has said she is willing to go on a waiting list. This happens all the time. The extrapolation that is probably being made is that the parent is trying to pressure the teacher by saying "I can wait as long is it takes." The extrapolation is also that this parent has absolute freedom, is totally able to take any other time, but insists on having her own way and will be equally inflexible later on. But do we know that? It could be true. It could also be true that this person has weighed everything carefully, this works, and the person will also be equally careful in making sure her child practises, has a supportive environment for practising etc.

I get customers calling me all the time because my work is not long term with anyone. I have to weigh things. Recently I got a call from someone who I thought would be a "difficult, pushy" customer. She turned out to be sweet, grateful, and accommodating, but had been given such a run around that she had become ultra cautious and defensive by the time she got to me. It taught me a lesson in prejudgment.

Meanwhile as student I have been at the other end of second guessing. We have no idea what teachers have encountered with other students, and that anything we say or do will remind them of these experiences, making them add meaning. It is a very frustrating thing at that end.

We really do not know, you are right. However, I've encountered enough of people who are demanding like this who end up treating me like I'm the "service/servant" rather than a highly educated professional hired to assist them in learning an instrument. I seem to recall that this is all through email, as well, and some people come across differently in email than they would over the phone. I always want to talk to the person on the phone and meet them before we even talk about schedules. That way I can really assess who I'm dealing with.