Hi there, we breed pedigree cats and have 2 children (oldest turns 3 in a few weeks and the other is 9 months).
My daughter gets to handle the kittens when they are 4 weeks old and a kitten we bred who shares her birthday with my son, sleeps on my daughters bed with her at night.
You may want to consider an older kitten from an established breeder (we sometimes keep them for evaluation) or a retired breeding queen/stud. Each one will be pretty stable in personality and very well socialised.
There are many different breeds out there, I would recommend this option over a rescue, especially with young children. I have a rescue cat, she has been with me for 6 years now and is the one most likely to hiss at the kids for pulling her tail, but then, she is 12 so it might be more to do with just being fed up of being old.
Some reasons for so many moggies and half pedigree cats in rescue centres is because people think they can get lots of money and there are people out there buying them. All of our cats leave here neutered (kittens who leave at 14 weeks are also done).
I can vouch for the wisdom of rehoming an ex stud/queen. We've just been given a 14 month old pedigree cat (can't believe our luck!) and he's perfect for us. We have 2 children, and i work every day, so a pre-litter trained, well socialised, known quantity is exactly what we needed. He's beautiful, intelligent, has settled in really quickly and i have the reassurance that the breeder will happily chat to me if i need any advice. We couldn't be happier!