Please keep my basenjis in your thoughts and cross your fingers that they get a fence, sooner rather than later.
Fingers crossed that it works out for you! I agree with everyone here that an invisible fence alone is inappropriate for a Basenji.
But with an invisible fence system you are not in control of timing, it is delivered when the dog gets too close to the boundary and if there happens to be people, pets, etc at the boundary then the shock can be associated with those things. This is even more likely if the reason the dog approaches the is because of those things.
As far as how often it happens, I think it happens far more often then you think based on the experience of trainers who are called in to do evaluations of dogs in homes with invisible fence. I know you don't care because it is not scientific research but it does exist.
Trainers called in to do such evaluations typically have been called as a last resort because the people are dealing with behavior they can't control. Whether or not the fence was a contributing factor, it is extremely likely that someone who gets into this position with their dog was not a competent trainer to begin with and didn't properly introduce the fence.
And yes, you are in control of the timing when you are training the dog. If the dog has been properly conditioned to the fence it understands that the effect of the fence is not dependent on whatever else is in the environment. And the dog will not blunder accidentally into the fence field. Invisible fence collars typically warn the dog when he is getting too close. There is an audible tone. After hearing the tone, the dog has the option of backing off or continuing. When you begin teaching the dog, you never let him feel a shock. You teach him that the tone signals he should retreat. Only once you have established this response do you allow him to get close enough to discover there are consequences to ignoring the tone. Most dogs figure out cause and effect very quickly, and never again encounter the shock because they are not stupid! You should also carry out extensive distraction training to "proof" the dog's obedience to the fence. Again, in circumstances you can control. And with maximum value distractions. Many people (most?) are not this diligent, even the so called professionals.
With some dogs, if there is an irresistible attraction perhaps they will continue, knowing full well what the consequences will be. Under those circumstances you are dealing with a dog that is so motivated to "get" the object of his desires that with or without the shock there is going to be a bite if he can get at it.
I care a lot, believe me. I don't think any sort of electronic training devices should be available to anyone without adequate understanding of their use, but unfortunately, as with many things, competence is not required to purchase them. This is unfortunate for both the reputation of the equipment and the unfortunate dogs. I think this equipment has its uses, but it needs to be respected and understood.
Often, we don't know enough about the circumstances which created an aggressive dog. Children are notorious for teasing, but once bitten their cry is always "I didn't do anything". A dog that has been harassed and finds an opportunity to retaliate usually will. And not only children, but adults also, will lie about the circumstances surrounding a bite. If they can blame some outside factor rather than their own actions, they will. I think this was quite clear in the story of the woman who was correcting her dog for digging, and blamed his aggression on the fence. The likelihood was that she made a mistake and triggered the response herself, especially since it took several approaches by her before the dog retaliated. It is extremely unlikely that he was digging in the radius of the fence field…...