Curly quotes are problematic on low PPI screens. They tend to just become unidentifiable clumps. The features that make them distinct become unidentifiable, they only look slightly odd and uneven.
I do, however, think that as we are moving to higher PPI screens curly quotes do look better (and, as you can see from my German example, some languages require you to put quotes in places where you cannot put straight quotes). I, for example, really don’t think straight quotes look any good in print which is high resolution. In print all the small features of quotes are always clearly identifiable.
I actually think that Georgia has nice curly quotes but they do clump up below 12px or so font sizes. I think no font is free of that problem. I still like to use curly quotes, well, just because. I certainly can understand why you wouldn’t want to use them at small font sizes on the screen.
(There is one exception. I like to use Letter Gothic for my snail mail correspondence. Letter Gothic is a great monospaced typewriter font and to preserve that typewriter vibe I like to use straight quotes.)