So the last version of Visual Studio that built MSI installers is 2010
I was using 2010 to build an installer for a 2012 Service
The MSI files were created, and they did copy the EXE files
However, the service install dialog did not occur
I did not do a step that I needed to do
On the menu for the installer project, there is a gear button
It allows you to invoke a custom action for each of the 4 states of an installer
(Install, Repair, Commit, Uninstall)
I had to add the service exe to each of those steps
Once I did that, voila, the service did what using installutil does on a development machine
Problem solved