My Vim setup has it (by default) so that non-i keys will enter insert mode. This includes a and s among others.
Is there a setting to make i exclusively enter insert mode?
No, there is no other way than remap all commands that start insert mode.
:h normal-index and :h visual-index as there is hell lot of different ways to enter insert mode (and one that you cannot disable :startinsert). However my main question would still be valid - why?