You have to add a client element in your updater xml:
Joomla seems to take the admin as the default.
I tried out your xml and found that joomla was indeed finding the xml file ok and even inserting a record into the #__updates table. However because your module is a site module and the update xml didn't have the client ...
Adding an answer since mine was similar to itoctopus with SiteGround. Instead of a plugin there was a service created by SiteGround that was causing the css file to be cached. It could be interacted with via the CPanel: