Extraditing from Sweden would probably require the permission of the UK government, who are generally more than willing to bend over backwards for the US on this kind of thing. What it avoids is getting the UK courts involved; how much that matters in practice is an interesting question. Also, I don't think the US was ready to extradite him yet back in 2010.
Edit: yep, arrest warrant supposedly issued in December 2017 over his work with Manning back in 2010.
It wouldn't avoid the UK courts getting involved; consent from the UK for onward extradition from Sweden is subject to judicial review, much like any other extradition request.
Edit: yep, arrest warrant supposedly issued in December 2017 over his work with Manning back in 2010.