This is a new one on me but I have just come across this Google advice for one of my keywords, So, is how often your ad is shown related to its QS? Or, is it just badly written (i.e. QS up = CPC down (theoretically!) = more ads for budget)? PS. I am running 'accelerated display' that I assumed put you in the auction for every search on your keyword until your budget is exhausted.
I read it as meaning that your advert will appear higher on average, for your bid. Since your advert appears in a number of positions for a search, the higher it is, the less likely it will be to fall onto page 2 for a search, so the less likely you are to lose impressions. Slightly ironic that in fact, if you are spending your budget, or close to it, then if improving your Quality Score improves your position, and hence your clickthrough rate, it will lead to you spending your budget more quickly, and getting fewer impressions! As you say, if you stay in the same position, your cost per click will fall, so your budget is more likely to last all day...
That could be it, Google do have an overriding obsession with page 1 placement! The fact that some of us may be very happy with our performance on page 2, 3 or 4 seems an alien concept to them. Yes, the issue of improving QS is a bit of a "double-edged sword" giving better position for your bid but also increasing ave. CPC and depleting the budget faster.