D&D 5E Handling Morale Checks

[Rune]

So, let me get this straight the rule presented in the DMG is...

If the monster is losing the battle and it FAILS the wisdom check it runs away.
So the wiser the monster is the more likely it is to stay and fight a losing battle?
That makes no sense...

Makes sense (that they chose Wisdom) if you consider that 5e assumes Wisdom is the stat that resists fear effects. Personally, I’d think it would be better represented with an independent stat (“Valor” or somesuch), but…eh.

Honestly, this is one of those things that the DM just decides on what makes sense.
Only things that should fight to the death are when there is no other option, the monster believes that if they surrender they will be killed anyway, the monster is in some sort of rage state of mind, or the monster is a fanatic (like a cult) for it's cause.

Well, DM discretion is the default when it comes to monster morale.