On all the agile projects I’ve worked on, the teams have been very reluctant to make a specification in place before starting development.

I don't think this is an Agile thing, at all. I mean, look at what Agile's main trait: multiple iterations with acceptance testing and product&design reviews. At each iteration there is planning. At each planning session you review/create tickets tracking goals and tasks. This makes it abundantly clear that Agile is based in your ability to plan for the long term but break/adapt progress into multiple short-term plans.