You think you vote for president? You dont - you vote for a slate of Electors who are committed to vote for your candidate for POTUS. Therefore, the only issue is that the Dems might have to get their voters to vote for Hillary as a figurehead. But they still face the issue of campaigning for those electors - and thus they have to convince their voters that the electors will vote for someone the voters like.
We know that many voters base their votes on the candidate's perceived personality; and if that personality is gone from the race, for whatever reason, there's a good chance his or her voters will stay home.