Dogs stay dogs. Children do not stay children. Dogs can expect to have a human around for essentially their entire lives to tell them what to do, so it makes sense that they are reliant on instructions. Children, even fairly young ones are sometimes expected to use their own judgement to make decisions. Dogs are not, because they are dogs. It's acceptable to leave a dog alone in a house, room, or even a small crate (as long as they are acclimatised to the crate and happy there) if you are unable to supervise them for a while. You cannot shut a child in the playroom while you go out, much less the kitchen.
Also, ABA is not clicker training, and is nothing like dog training. This is linked on the petition but probably lost in the multitude of much more wordy links - worth reading. A statement from a professional dog training association: www.pact-dogs.com/blog_post/pact-official-statement-regarding-channel-4s-upcoming-train-baby-like-dog-programme/
I do think there are some aspects of dog training that work well in child discipline - the concept of incompatible behaviours, rewarding and marking the positive, consistency, recognising and making small steps, controlling the environment, the "training bubble", making sure what you're asking them is clear in "toddler language", understanding what the child needs to behave well. Unfortunately none of these things seem to be what the programme is focusing on
meaning it's simply a sensationalist thing probably intended to be controversial - I expect they are delighted there is a petition!