If you've come to realise this over the last couple of weeks, that suggests you are still in the very early days of understanding what is happening for your dog.
I am guessing, but just to be sure, you have had a recent vet check (i.e. since he snarled at your child or barked at your Dad)? These behaviours could be triggered by something medical, in spaniels the obvious suggestions are something like a very painful ear infection, making their tolerance much lower than usual. That's worth doing and probably somethingyour behaviourist has already insisted on.
At 4 (and possibly 7), your child ideally should not be left alone with any dog, but of course one with a history of aggression means the risk of getting this wrong is even greater.
Assuming your behaviourist has not only talked to you about training, but also ways to arrange your home to keep everyone safe while you make a start with the dog, I would keep going with that for a little while - but not too long.
Safety is parament, even more important that the dog's welfare (and I don't say that lightly) so if your dog has to spend more time than ideal, secure in a room by themseleves, maybe behind a baby gate or similar - as a TEMPORARY measure - then that's what has to happen to keep everyone safe.
Living for ten years on constant vigilence is very hard indeed. Through that time your children will grow up and want to have friends over etc. That's a lot to ask of your whole family and is a lot to ask of the dog.
Without a medical cause and without significant improvement in the short term, I would consider rehoming. I've known SpanielAid (as mentioned above) to rehome reactive dogs and often they do much better in quieter, rural, adult only homes. However, they have the best shot at this is they have not got a history of biting, which is another reason why you don't have months and months to make the decison.
It's a shit hand, OP. I've seen the heartbreak caused when the dog's personality and what is needed by their family are so mismatched like this. Reactivity is a complex business and multi factoral so I wouldn't beat yourself up over neutering etc. There will be much more than that playing a hand here.
(That said: this is one of the reasons I am not keen on neutering at such a young age - these behaviours often don't boil up until a dog is 1-2 years old and if you've neutered early, you don't get a chance to discover them until its too late. Not a criticism, just food for thought for any future dog)