Vedanta uses 'Nirguna' (without qualities) and 'Gunatheetha' (beyond qualities) as synonyms. In a practical sense nirguna means imperciptible. But, if we don't perceive something, how can we be sure that it exists, right?
Well, vedanta has a different opinion. Wisemen say anything that exists can not be perceived, and anything that is perceived does not exist. For example, you know that you exist. How do you perceive that? Obviously you know that you exist, that is, you experience that you exist rather than perceiving. However existence of many other things is perception for you. You don't experience that. It is for this very reason that vedanta teaches to trust on what you experience rather than what you perceive. The experience of self tops them all. This is why you have to find Brahman in the feeling of self first.
Source: "Who Am I" by Ramana Maharshi
(I agree that it is not the kind of source you expect. But this is the source I used.)
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An answer to the question in the title.
Answer to: How can Advaita Vedanta's Nirguna Brahman even exist?
You are saying that only objects having quality exists. This is arguable.
But we are all clear about one thing: the reverse direction. And that feels more natural.
Only what exists has (or does not have) quality.
How about that? And another related question. Are you sure 'qualtity' exists? If so, what is the quality that makes it exist?
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