For a thought experiment: Let's say you could hit the really optimal headspace, for perfect vacuum. In this case, during the cooking phase, the food will expand such that it fills the jar exactly, pushing out all the air. When you cool the jars down, there will be no air sucked back in (the jars are designed in a way to prevent it) and you'll have completely airless vacuum inside the headspace. If you leave too much headspace, some of the air will stay in the jar without getting pushed out, and you'll get a less hard vacuum.
As some anecdotal evidence that this happens, I once pressure-canned some stuff in small old-fashioned Weck jars (they have glass covers with separate rubber seals). I had too little headspace, and liquid from the jar leaked into the canner. It didn't impair the seal in this case, and I ended up with very strongly sealed jars. They're difficult to open, and the gaskets are so deformed that they can't be reused, even for non-canning closure.
The classic safety upside of having stronger vacuum will be that you're better protected from potential spoilage through aerobic bacteria. In principle, following proper canning procedure will leave you with no viable bacteria at all, and in the cases it fails, you'll notice it when the seal fails (or the twist-off lid bulges). But of course, this has only been tested with proper headspace, not with half-full jars, so nominally, you have to follow the recipe to the letter to claim safety.
Also, air leads to discoloration of your food. The less air you have left, the less your food oxidizes, so you keep better quality.