Hi Andy, remember me - I sat with you in the Dakota the first time you tried it.
You can pretty much choose your own level of risk in a private aeroplane flight. You can choose to take additional risk by flying over water, or at night, or over terrain which wouldn't be friendly if the engine failed, or you can choose not to. You can choose to fly in bad visibility under a low cloud base, or you can choose not to. I, personally, sometimes choose the additional risk of flying a single at night or over water and will have as a passenger any adult who I consider to be capable of understanding the increased risk if I explain it to them. When I fly with my son, I don't consider him capable of understanding the increased risk and so I have to understand it on his behalf - because of that I won't choose to take any additional risk with him in the plane.
In any flight, there is a small element of risk which is beyond my control, but I wouldn't like to say that this is less or more than the risk of taking my son in the car - which I and most other people do without thinking about. My son only flies with me occasionally, but he's in the car or walking next to a main road regularly, so my own perception is that the increased risk to him of flying is insignificant.
You'll just have to agree with Mrs. Andy that any risk to your children are acceptable, but as far as taking her along too, don't make her go if she doesn't want to - the motivation should come from her. I once made the mistake of encouraging my (now ex) wife to sit in the back of my microlight shortly after I got my licence to fly it. She reluctantly agreed but, immediately after take-off, she got all hysterical and convinced herself that she was going to die. I flew the shortest circuit to land that I could but I wished that I never encouraged her in the first place.
I suggest that if you have any doubts about your children's ability to behave and not distract you with their over-excitedness, that you take someone with you who you know to be comfortable in the aeroplane and you trust to keep the children under control while you concentrate on looking after the flight. Once you know that they are OK you can dispense with that person.
If your wife is nervous then she might not be the best person to do this for the first flight - you might end up looking after the entire family while you fly the plane. Alternatively, having to look after the children might take her mind off being nervous. You know her and I don't so you'll have to decide that one.