Here is the solution from an ESRI support rep. The reason why you are not able to remove the original ESRI basemap is because it was design that way, when you open a blank web map. But below are the steps to remove the ESRI basemap.
Method 1 :
Step 1: Copy the basemap service URL.
Step 2: Open a new web map
Step 3:Add layer from Web.
Step 4: Check "Use as basemap"
Then the "Remove" option will be available for you to remove the original ESRI basemap.
Method 2 :
Step 1: Copy the service URL.
Step 2: Open a new web map > Add > Search for layers > click on required basemap > "Use as basemap"