Let me preface this opinion by saying I’ve had 3 JRTs in Ubersuburbia, so different challenges than a farm life. I’ve raised 2 from pups, 1 was a rescue.
Given your situation-new to JRTs, 2 very young children-- if you were to pursue the breed, I’d counsel a puppy from a reputable breeder you can visit, to see and evaluate the parents. No one knows the JRT better than (reputable) breeders, and (reputable) breeders will honestly answer your Q’s as to appropriateness.
I dont think an adult rescue would be fair to you or the dog, unless you got very lucky. IMO, JRT rescues need experienced JRT/terrier/rescue dog people. JRTs in general aren’t for everyone, and as much as I believe in adoption, having done it many times, I think you have to own what you may be bringing home with a rescue Jack-possible animal aggression, territoriality, high prey drive, high reactivity, and YUP, YAPPING.
When I adopted my guy, a 1+ yr.o. male, he was fine with other dogs, friendly, and low man on the totem pole in the group of small breed rescues the rescue lady had in her house. PERFECT! But he came with other issues that I feel quite certain he wouldn’t have if I’d been able to get him as a young pup. We love him, though:)
I’m curious what makes you believe that a rescue of any breed would come “already trained”? IMO, you can NEVER expect a rescue to be “trained” in anything. When a dog walks into your house, its a new beginning and its up to you to teach him how life works in your house. “Expect” anything more and you will be disappointed. If you go the rescue route with any dog, go in with eyes wide open.
I love my Jacks, but I wouldn’t put my rescue guy around young playing children–he’s very reactive, and ramps up quickly. My girl is much less so and even at just a year, is fine around even young kids running around-never gets too excited. Nature or nurture? I’ll just say that the 2 I raised from puppyhood were both fine with kids-perfect dogs, really:)-as are the 2 my sister has, and I wish I’d gotten my rescue guy as a baby.