That division is in fact not universal and not clear.

First, in some armies promotion of enlisted men and NCOs is or was in fact main source of officers. One example is German army before and during WWII. Link in Russian

In short, no higher education was required and training itself took 6 - 9 months.

Second, USA has warrant officers, USSR had (and some post-Soviet states now have) praporshchiks, and there are many more categories that are between enlisted and officers.

And reason why that two-ladder system with classist origins is not univesally abandoned as obsolete is simple: long education. To promote senior sergeant to leutenant you have first sent him to academy, and if that takes few years that is not really an option. So choice is between educated and experienced officers, and most armies choose educated.