How to salvage hardened peanut butter from jar bottom

The answer is mostly just "it's tradition", as with most questions like this.

I do think the pattern you've described isn't quite the actual one. What really happens is that we tend to eat peanut butter and jelly on sandwiches, and put one or the other on single slices of bread, because with a sandwich you can spread one thing on each half and put them together, but with a single slice of bread it's messy. Sandwiches might be on toasted bread, though - untoasted is common and easier but plenty of people like toasted bread for sandwiches. Similarly, there are certainly lazy people out there who'll just spread something on a slice of untoasted bread.

Beyond that, it's just what we do.

There's definitely some rounding going on because the peanut butter has 100.1g of nutrients per 100g of product. This isn't enough to explain the discrepancy. Adding up the nutrients on the roasted peanuts gives 95.4g. I think we can assume the other 4.6% is water. So perhaps more water has been driven off the peanut butter.

What I think is more likely (though could be additional) is that there's significantly less fat in the peanut butter (48.1% vs 51.7%). While this could be down to different varieties of peanut, I suspect some fat is removed in the processing, perhaps to avoid an oil slick on the surface.

Most likely of all is just different source data. Different peanuts may have been used in the calculations -- whether that is consistent with the actual ingredients used is another matter. In fact if you look at nutrition information for raw peanuts online, you'll get a range of values. There's nothing special about peanuts, the same is true for bananas. I'm sure if you look at the scientific literature on any foodstuff, you'll find a range of values published, reflecting natural variation as well as measurement variabilility -- this will then propagate to the values published to the consumer.