Here is a framework for how to ensure that WebDrive maps properly for SharePoint Online connections using the REST API connector option when rooting the drive on a specific subfolder.
1. Go through the process of setting up your SharePoint Online site using the REST API connector option. Setup the connection to the highest possible folder that the user has permissions to. For example, if your desired folder destination is https://company.sharepoint.com/sites/sitename/Documents/subfolder/etc -- setup the initial connection to https://company.sharepoint.com/sites/sitename
Skip over any advanced settings or options during this initial setup to keep everything clean and consistent. Simply choose a site name and drive letter when prompted and then save the site. You can also choose not to map the site now (just Finish), but choosing to connect now will not hurt anything - you'll simply need to disconnect in order to do the next step of enabling Root Drive on this folder. The site should not be connected before going to setup the Root Drive on this folder setting.
2. With the site disconnected, go to Site Properties --> SharePoint Settings and then click the checkbox to enable "Root Drive on this folder" and then click on "Select Folder". Here, navigate the tree of folders to select the proper subfolder. Select the folder, and Click OK and then OK to save the changes in WebDrive.
3. You'll notice at this point that the site is connected, as it is required for the site to map in order to find and choose the root folder. Disconnect the mapped site.
4. Go to App Settings --> Cache Settings and choose to "Clear Cache". This is necessary in order to ensure a proper connect to the specified root folder on the next connection.
5. Connect the site in WebDrive and it will properly map the drive and open to the specified Root Folder.