Thanks for all the suggestions. After looking around, the Rivendell bikes seem to be designed with this in mind: road riding in a comfortable, upright-ish position. They have long chainstays and shallow seat tube angles. The low bottom bracket is appealing too.
Are there any other builders taking a similar approach to their designs?
If not, it seems vintage may be the only affordable way to go (e.g. older Schwinns).