To expect to be able to re-home a rescue dog when I have two children?

[Wadingthroughshit]

Two children, We live in a rural area, have front and back gardens, my youngest starts school this summer, my eldest is nearly nine and I work PT. There are kennels are dog walkers in our area which are reasonably priced. We are a quiet home, there isn't much comings and goings. I think a dog would be hugely beneficial to all of us, especially my eldest who can be slightly anxious.
But, I have been looking for over a year, and I have not been able find dogs at any of the rescue centre which don't don't state 'adult only or older children'. I absolutely understand there is a reason for this. Should I give up looking and give in to buying one? We'd quite like an older dog, well, not a puppy at least.

In terms of fences, we made ours high enough by adding trellises which was easy and cheap. We had exactly the same experience with the big rehoming charities (my youngest is 7) and in the end rehomed an ex-racing greyhound who we all adore. He’s very soppy and affectionate, loves games in the garden with the children and adjusted to life in a house v quickly. He got housetraining immediately and we’re working on recall. He is still nervous but was in kennels for nearly 6 years and has only been with us for a month, already he’s much more playful.

So happy to see so many recommending lurchers! Here's my baby, adopted from the RSPCA after being used for badger baiting and sustaining awful injuries from it (he was given lifesaving reconstructive surgery when he was rescued by the RSPCA, about 2 months before we adopted him). He's the sweetest, most loving boy I've ever met and all he wants to do is cuddle.

Back garden has fences, front has stone wall, a very quiet road, then big field opposite. Oh dear so this may be another obstacle?

How high are the back garden fences? Dont worry about the front, as long as the back garden is secure thats good enough. No reason really for a dog to need to use the front garden if the back is secure.
Love seeing all the greyhound and lurcher rescues. Our boy is an ex-racer and he is our first dog and we love him. They are a very quirky breed which just makes them that much better!