Watermelons, while that method may be common, there are other ways to euthanize via inhalation which are listed in the AVMA guidelines I posted above. I don't know what technique this particular veterinarian is planning to use. In some animals who are that ill, it becomes incredibly difficult to hit a vein and if circulation is poor, that technique may not be successful.
Your vet should be able to explain to you his or her reasoning for any technique or drug that is used. If they are unwilling or unable to do so, go elsewhere.
I personally prefer when the owner stays with the pet during a euthanasia. I've seen too many of the classic "we got a new puppy and we're tired of the old dog so we dropped it off to be euthanized and went home to play with the puppy" sort of euthanasias. In those situations, at the clinic where I worked, we always made sure that one of us was with the pet holding it and petting it and making sure that it knew that someone cared in its last moments.
I suspect that your vet may not have the ability to bring the sedative gas into an exam room (which is typically where euthanasias are performed with the owner present) and may not be comfortable bringing an owner into the back of the hospital (it can be a liability issue).
With regards to having children in the room, some animals will vocalize, urinate and defecate, and have muscle tremors following euthanasia. This can be very confusing and distressing for many children (and adults) and I think it's important that all parties in the room at the time are prepared ahead of time for what may happen when the animal is euthanized.
Like I mentioned above, talk to your vet about why they've chosen their technique and if it's important to you to be with your pet, you should be able to make that happen.
