YANBU to have a dog with young children.
You ARE being unreasonable to write off an older, assessed dog from a reputable rescue.
You ARE being idealistic in taking on a puppy given your current commitments for all the reasons Dooin has stated and more, unless you have a good amount of experience and/or the type of commitment which does not come with the average family or individual.
You don't come into that category of exceptional experience or commitment as your remark about taking a puppy jogging
demonstrates.
Am I right in suspecting that you have never owned a dog or puppy of your own before?
You ARE being incredibly unreasonable to buy a puppy when thousands of healthy young dogs and puppies are being killed in pounds across the country each year purely because they have no homes.
I'm sure you'll disagree with me, but answer me this:
Why do you think that so many rescues will not rehome to families with young children and why do you think that those which MIGHT do so are generally so very selective even if they don't have a "no young children" policy?
I'm a rescuer. At the moment and on the strenghth of what you have said here I wouldn't rehome to you as I don't think you have enough knowledge yet and because you don't appear to be suitably experienced - therefore I don't honestly think that the homing of a dog into your family would last.