First, replacing the vice president is almost always a bad play.
The last president to pull it off and win re-election was FDR. Eisenhower, Nixon, and Bush Senior (or at least some of their advisors) all toyed with doing it, but there were constituencies in the party that rallied behind the VP. The one president to actually remove his vice president and replace him in the past half-century was Gerald Ford. Replacing Rockefeller with Sen. Dole (who came off badly in the debates with Walter Mondale) probably wasn't what defeated the ticket but it certainly didn't help.
On the off-chance this could be engineered, Hillary would not be a good replacement. This is not due to any fault of her own. She has proven savvy and capable as secretary of state and her experience as an active first lady legitimately gives her insight into the unique pressure cooker that is the White House. However, she brings her husband and he is problematic baggage. Again, this is not through any fault of his own, but rather because ex-presidents need to be kept a healthy distance from the Oval Office. Bush Senior was rarely around when his son was president and Reagan stayed away from Bush Senior. These were sound precedents and should continue.
A few additional notes. First back in the nineties I was a bit of a Clinton-hater, I have come to respect the virtues (irony alert) of the Clinton Administration. But I still think ex-Presidents need to stay off the main-stage. That being said, nothing prepares one for the Presidency like serious time in the White House.
Biden's comments were pretty bad, and if it were a Republican probably would have far more blowback. One of the difficult things about politics is how policy preferences and values began confused in the public debate. Opposition to a specific policy intended to help a community or population does not mean those opposed hate that population. It means they have legitimate questions about the efficacy of that policy or the whether that policy should be prioritized. That is not to say that this opposition is not at times a cover for unacceptable views but one should not leap to that conclusion. Being a Republican does not mean I don't care about the (fill in the blank here - poor, women, African-Americans.) It means I have questions as to whether the policies Democrats prefer will help that population or whether the other costs of that policy will outweigh the benefits.