Present Simple or Hard Present ?

'The sun rises in the east' -
a commonly used example of the Present Simple
Photo by @CliveSir via ELTpics on Flickr

In a recent discussion in one of the Facebook groups (this is what seems to prompt my occasional posts these days), the Present Simple was referred to as 'one of the hardest tenses for students to get'. This made me wonder whether the Present Simple, contrary to what its name suggests, is indeed not so simple, or it is just another one of those teacher-induced neuroses. Let's see why there's so much ado about the most common, unmarked English tense.