Your judgement is absolute. But you realised this later.
An interpretation is not bending the meaning of something; it is what you genuinely understand it to mean.
Think of it like translating a passage from a foreign language. Different people read this other language to mean different things. Hence when one person translates it, the corresponding passage may be specific in some regard, but another may not. While the interpretation I'm talking about is not this extreme, it is not far off. Everyone has a different understanding of the English language; everyone interprets things differently. If in someone's interpretation something is stipulated, then to them it is stipulated. You can't tell them that they are wrong, there are similar biases in your own interpretation.
Bottom line: your interpretation is not necessarily the "correct" one.
Based on very superficial knowledge, I made a logical assumption.
Hmmm, that doesn't seem right. Read that back to yourself. Again.
While that may help you for this debate, you have absolutely no idea if there really is no mandate in the Quran. But, alas, you still go ahead and state: "It is not specifically mandated!!" And all on superficial knowledge! This is akin to the childish attitude, "It must be true; I read it on the Internet."
My point here is don't be absolute. You don't know that there is not a mandate in the Quran, and even if you did not find such a passage, does not mean that someone else will not. You are not an authority on this, and neither are the top 8 results of an internet search. You can't make a logical assumption and say that there is no textual base for someone's beliefs, based on the words of several anti-Burqa websites.
My sole purpose here has been to show that there is the possibility that there might in some small place in the Quran be some basis for this tradition. I don't know, and neither do you. So don't go out making absolute judgement.