Does the artifact feature accomplish this?

My build routine creates a file called "compile.txt" in the root folder  (my build script is in $source.myrepo$\build\mybuild.fbp7)

I would like that file to be "logged" with CCI so that I can see that file for each build.  

Is this what artifacts are designed to do?

Yes, you need to define a Workspace Rule to get the file copied back to the server workspace, and then an Artifact Rule to have it listed on the Artifacts tab.