Time of a story in a stage
It should be possible to see both the total time a story has been there but also how long time a story has been in a specific stage.
In the desktop version you can see the lead time (number fo days from creation to completion). Kanban you try to optimize the average lead time and thats the reason why it is presented.
When I first started working on Kanbana I had a lot of ideas on how to help users become more effective by finding bottlenecks and so on. This requires some of the data that you talk about. But after running the http://survey.io/ customer development survey I found out that the top most reason for using Kanbana had nothing to do with productivity but rather clarity and balance.
The next step after gaining clarity and balance is effectiveness but as it is right now I don't want to scare people with to much details that has to do with effectiveness.
But I actually track both leadtime, cycletime and stagetime so it is actually possible for me to add the features you talk about, but I want the users to be ready for it before I add these features.