The second part of your post is correct, but you're misunderstanding the meaning of "elasticity" as it pertains to metals. Short answer: Annealing reduces elasticity.
Long answer: Most metals, including brass, are "elastic" to one degree or another. It might be easier to think of a piece of steel, as an example. A piece of "spring" steel, like a piece of "piano wire" is made from tempered high-carbon steel. It is a perfect illustration of a metal having high elasticity. You can bend it quite a bit and it will spring back to the original shape. However, if you bend it far enough, it will not spring back. Those who study metals would say you "have exceeded the elastic limit" of the material. If you take a piece of a wire coat hanger or straighten out a paper clip you can bend it a little bit and it will spring back to the original shape. But, since these metals are not tempered in the same way as piano wire, it takes less bending to deform them; i.e. they are less elastic. When you bend most familiar metals enough to permanently deform them, you are in the "plastic deformation" range.
Cartridge brass, when it's fired or sized, undergoes plastic deformation and we can easily see that by, to use one example, measuring the neck diameter before firing, after firing, and after sizing. Repeating this process (without annealing) makes the brass harder, more "springy", more elastic. This process is called work hardening. The brass becomes more brittle and subject to cracking. It also "springs back" more after sizing because it is more elastic.
Annealing cartridge brass reverses the effects of work hardening and makes it "softer". It becomes more "ductile", meaning it's easier to permanently deform. It is correct to say that it becomes less elastic after annealing, not more elastic.
I suspect you already know all this with the exception of how the word "elastic" is used to describe the behavior of metals.
Yes, I was confusing the terms and appreciate the tutorial. It should be helpful to more than just me.
Rick