The problem with the current functionality is it leads to misquotes. I’ve multiple times had people post a quote on my forum that says I wrote text, but the text is actually from some book I quoted. If the quote level was correct, then people would see the attribution wasn’t quoted rather than thinking I wrote the text.
I think it should be safe for non-technical, non-precise users to select any text, click quote, and get a quote that isn’t clearly wrong.
Email lists handle this better. Observe me selecting quote level 3 text and then hitting reply in Mac Mail:
Expecting users to always quote the attribution in addition to the quote doesn’t work well. Users may not know to do that. They may not want to clutter up their post with extra text. They may be only quoting the middle of the quote, so to get it right they’d have to manually edit out text in between the attribution and the part they’re quoting and type an ellipsis. They may be quoting multiple sections of the quote to reply to separately but they wouldn’t want to include attribution multiple times in their post.
I do think there are also issues when you e.g. quote a phrase that was bolded and get a non-bold quote because you didn’t quote the start or end of the bolded text. But this tends to be much more visually obvious to posters in the preview than a missing quote level inside a blockquote, and the harm of errors here is smaller than with incorrect quotation attributions.
I think that however you handle this stuff in general, quote levels and attributions in particular should never be wrong when users don’t manually screw them up.