This is not a perfect solution, but you can use the SO search, rather than combining the unanswered and featured tab to get part of the way there.
The reason it is not a perfect solution is there is no
hasbounty operator in the advanced search, but there is a
hasnotice, where the notice would be any notice, including the bounty notice.
answers:0 hasnotice:1 locked:0 is:question lastactive:2013-10-05..2013-10-12
answers:0 (filter on questions that do not have any answers)1
hasnotice:1 (bounty is considered a post notice, but there are more kinds of notices)
is:question (show only questions, we don't want stuff like locked answers)
locked:0 (exclude most common notice that isn't a bounty)
lastactive:[week ago date]..[today] (format yyyy-mm-dd to exclude questions with notice like "Insufficient explanation" and others)
My original query (
answers:0 hasnotice:1) returns 112 posts, but the amended one from Sha Wiz Dow Ard, who identified some some more advanced search criteria to help filter on Featured Questions more accurately, trims this list down to 104 posts.
A major disadvantage is that the system classifies posts with unupvoted answers (with none being accepted) as unanswered as well, and this will not return these. Searching for posts by answer score is not implemented, but you can do another search with
isaccepted:0 instead of
answers:0 to find these, but they will be any unaccepted post, not just those without upvotes.
There is a feature request, Advanced search criteria for bounty questions?, to add
isfeatured to the search operators so you don't need so many operators to filter on bounty questions. So you can show your support for that post, but it still won't correct the limitation with unaccepted 0-score posts.