Non-magical shields do not take up a magic item slot yes. Non-magical shields do however, take up the space on your arm that the bracer goes. And wearing the shield means you cannot strap the shield on, the bracer gets in the way.
Could you point to a page number where they might say something remotely like that? Page 224 in Players Handbook on how item slots work says the exact opposite - that you can, in fact, specifically wear a magical bracers and magical shield at the same time as the example. The problem being that only the magical item you put on first will work. This is not fluff text, these are the actual item slot rules.
As the shield is not a magic item, you don't care if its magical powers work or not...
Its the same reason you cannot wear magical or non-magical cloth under plate mail and expect to benefit from boots of striding and springing(even if you are technically "wearing light armor"), or expect to get the enhancement bonus from the cloth and the armor bonus of the plate. (I.E. trade a few points of AC for the enhancements you can put on cloth, but not plate)
Not at all the same thing. You can't wear two helmets at once(again, from PHB page 224), so no two sets of armor. You can wear a shield and a bracer at the same time, that's not a physical limitation.
Not to mention, neither of those would work well for other mechanical reasons.