No, you don't have to install Crystal Reports Designer on those machines that will not be used for designing reports. erwin Report Viewer (which employs a Crystal Reports runtime engine) launches when running the .rpt reports located at Tools - Pinned Reports.
In order to create new .rpt files, however, you do need to have Crystal Reports designer installed, but for those systems which merely run the reports, it is not necessary as the run time engine is automatically installed and has a very small footprint.