The thing is that Upload.aspx seems to be never used in reality as SharePoint will keep redirecting you to the _layouts/Upload.aspx page, which in its turn can be edited as well, but resides on the server and changes to it would affect the whole farm. This post has more details on why this is happening.
I've spent some time searching for a solution and the easiest I've found was to amend site's CSS to hide unnecessary controls.
All you have to do is to add the following lines into the file:
display: none !important;
The cool thing is that this will not only remove the Upload multiple files link from the upload page, but will also disable the feature within a document library. And it applies on a site level only, so you can leave the rest of your farm intact.
You can achieve the same by other means, but they all about code and amending server files which is not usually acceptable on a large farm.