BY RAW: Feats allow you to bypass restrictions put in place by general rules. Or to put it more directly, specific trumps general.
This question also relates to this: Feat Prerequisites
In this case, War Caster does indeed bypass the requirement for shield proficiency because it explicitly states that you can cast a spell with when your hands are occupied by weapons or shields. Of note, you do not get the benefits that would be granted with proficiency with shields. Which means you still suffer disadvantage on your ability checks, saves and attack rolls involving Str or Dex. It doesn't get more clear than that. The wording is very specific in that is states:
...perform the somatic components of spells even when you have weapons or a shield in one or both hands. (PHB p.170)
What's more important is the War Caster specifically states the pre-requisites for it: The ability to cast at least one spell.
Another feat which demonstrates that feats do indeed call out for proficiency when it is required is Crossbow Expert. Example (emphasis mine):
You ignore the loading quality of crossbows with
which you are proficient. (PHB p.165)
This example is often repeated. Dual Wielder bypasses the general restriction of light weapons. Magic Initiate gives you the ability to cast spells, even if your class doesn't have the spellcasting feature. This keeps going. Feats are exceptions to the general rule, and do exactly what they say they do.
Shield proficiency is not listed as a pre-requisite, nor is it mentioned even once in the feat, and therefore is not required.