Hi Kate, I teach piano and here are my opinions:
I wouldn't take a child who had nothing to practise on at all. Imagine trying to teach your child to tie their shoelaces or use a potty or a spoon but only letting them have the shoe, potty or spoon once a week for half an hour. It would take forever. Rent a piano if you don't want to commit to buying one. If you then buy, you usually get all your rental fee off the purchase. If you buy a reasonably new one, it will hold it's value quite well.
Kate, you could look into learning with your child. I will have 2 parent and daughter combos next year, which I'm really looking forward to.
I would agree with Freckle that choosing the right lesson time is vital. I give my pupils lesson times based on their age, rather than ability. A very young child will have more success right after school or on a Saturday morning.
Reading music comes easy to children, it is a cross between learning a language and a code, which children are easily able to do. Teachers use different methods to do this, mine used colours when I was little. I use music to teach the alphabet (we make little words with the notes etc. and then these become pieces)
Whilst I also agree with the points made about pushing children, a gentle reminder to practise and help building practise into routine (5 minutes before the TV can go on works well)is good. It shows you are interested, and gives the lessons a value. The students who enjoy playing the piano the most are those whose parents sit pretending to read a magazine on my sofa, but who are actually listening to every word ready to help and encourage at home.
Zebra, I know a lot of piano teachers don't travel because they usually have good instruments to teach on and all the resources at home. I would offer a Saturday morning lesson to you. I do have one pupil coming next year who is an friend's daughter and so I am teaching her after school whilst her mum looks after ds in the next room and charging a bit less because I don't have to put Bob into preschool. As mum would be there anyway it's a win win situation.
Some parents take their other children to the park, do their veggie shopping at a local market garden or play tennis on the court near my house. As the younger sibling get older they can sit in on the lessons to find out if they want to learn.