"why can't I see it in the list itself and add it to a view?" That has always been a great question!
You can create a custom code Event Receiver that gets the folder name and copies it to a text column.
I don't have access today to a SP2010 farm to test this, but I think you can also get the folder name from a SharePoint Designer workflow. It also would copy the folder name to a text column.
You should also be able to customize the view from SharePoint Designer and add the column to the view's web part. If not a displayed option, it can be added by editing the XLST code.
Steps using SharePoint Designer 2010 to add "Path" as a column.
- Create a view with the "Show all items without folders" options selected. (Not required to show the Path, but I think that's what you want.)
- Open SharePoint Designer 2010 and the Site.
- Click Lists and Libraries and click the library.
- On the right, click the View's name
- At the bottom of SPD make sure "Design" and not "Code" is selected.
- Click in the web part.
- In the ribbon click "Options" and then "Add/Remove Columns".
- Add the "Path" column.
- At the top left of SPD click the save button (disk).
- Test in your browser.
You could also add a Calculated Column to remove the site and library parts of the Path.