So, first, let's look at what you're doing when you trim asparagus. Like most other green "stalk" vegetables, as asparagus gets larger and older, the stalks get more fibrous as a way of supporting the plant. In addition to asparagus, this is true of broccoli, kale, and many other vegetables. One way to avoid this fibrousness is to eat very young "baby" plants. For example, if you simply buy asparagus that are pencil-thin or smaller, you often don't need to trim them at all.
If you do need to trim, though, Serious Eats covers techniques in some detail. The "snap at the natural breaking point" thing for asparagus is a pervasive cooking myth. It doesn't hold up under testing, and tends to result in removing much more of the asparagus than just the fibrous portion ... up to 50% according to Cook's Illustrated.
I trim my asparagus with a knife. In my personal experience, the best indicator of where the fibrous portion ends is to look at the color of the asparagus; the fibrous portion is usually paler, shading to white at the bottom. This can mean trimming some individual stalks separately.
In other words: your mom is one-up on Alton Brown, here.
As both you and Cook's Illustrated note, there's an alternative, which is to peel the asparagus. The inedible fibers form mostly in the skin and outer flesh of the asparagus, while the core of the stalk remains tender, just as it does in broccoli. And just like broccoli, you can remove these fibers with a sharp peeler or paring knife and still cook and eat the core. Personally, I rarely do this because it's a lot of work and here on the West Coast of the US asparagus is pretty affordable. But it's very common, even standard, in Europe.