Hello, guitarists

I'm starting to learn guitar, and one thing is bothering me. Cannot find explanation or similar question anywhere.

So there are basic chords - Am, Em, C. They teach that finger placement for these chords looks like this:

It seems to me this is much easier when switching between chords: to move from Am to Em you have only to change third finger position and remove first finger, and to move from Em to C you just have to put third finger one fret higher and first finger back as in Am

Is there anything I'm missing? Why don't they teach Em with second and third finger instead first and second?