The grammarians are basically correct. One should not say "quite unique".
However, consider the question this way.
"Unique" can be a relative term, one capable of being modified, if one gets more, not less, pedantic.
After all, everything in the world is unique. Even "identical" things are distinguished by iteration, version, or context. So, as the basis for comparisons shift, a thing can become more or less unique.
For example, say you have a room with two kittens in it, a black tuxedo kitty and an orange tabby. Each is unique. Their difference is pronounced and obvious.
Then, if you filled the room with 200 kittens of various colors, the criteria for "unique" changes. Each kitten is still unique, but now the group is more homogenous, and each kitten seems less unique.
But let's say 199 of the kittens were black, and one were orange, you would then have 200 unique kittens but one would be particularly unique.
This is the meaning of the grammarian's allowance for the sentence "Chicago is no less unique an American city than New York or San Francisco". New York, San Francisco, and Chicago are all unique, but in the wider comparison (all American cities) they are similar in that they stand apart. The unstated assertion is that other American cities are not unique, which is not strictly true. The use of "unique" is therefore already a relative matter subject to the shifting conventions of discourse.
Still, the best answer is not to say "quite unique". More so because "quite" is generally thought to be a sloppy cliche, a placeholder that weakens the sentence.