I found the answer to my question in the core.js code (i.e., the JavaScript code that's loaded for every SharePoint page). In the function AddSendSubMenu, the section that displays the "Other Location" option contains the following conditional statement:
if (currentItemProgId !="SharePoint.WebPartPage.Document"
&& (serverFileRedirect==null || serverFileRedirect=="" || HasRights(0x0, 0x20))
&& strExtension !="aspx")
Which basically says that it will only display the "Other Location" or custom send to destination if the item is not a web part page and doesn't use the "aspx" extension.
So I guess my question is now... WHY?
And and also, how should I do this? What's a good way to easily send a web part page from one document library to another? My goal here is to keep the working drafts of a web page in a collaborative site, where a limited number of people will have access to review the page and comment on it; then when the page is ready to publish, easily send it to a destination library on a publishing site where everybody in the company can view it.