IIRC the writer of the episode himself(?) declared that he wouldn't be surprised if what the probe showed Picard was a somewhat idealized version of events. Even if that sounds a little too sinister, it's reasonable to assume the probe did some customization of what Picard experienced to make it more relatable to him. There's also the fact that the experience seemed to be based around decisions Picard made, not a situation where he was a passive observer. I suppose taken to its farthest extent one might conclude that the real Kamin never even had kids, but that the probe whipped that up for our captain.