If AppCompatActivity and the standard library handle the usage of newer APIs in previous versions of Android, why do you need to specify a minimum SDK level?
Because the support library doesn't fix everything. It can't fix some things- there are features that require hardware support. Or are too big to backport. For those you need a way to specify what versions of the SDK aren't supported, hence the minimum SDK level.
Also the framework DOESN'T handle calling newer functions on older SDK versions. If you call a function added in v26 on v24, you'll crash.