Hi,
Indeed, it works only when the product is published/unpublished from the product edition page. I didn't think to check on the listing. I've made a patch so that this can work on the listing too. The patch is now on the current build available on our website and will also be on the next release.
For the rest, they indeed aren't synchronized. That's not really possible as that would require some dynamic trigger from the UCM mechanism when displaying the results page on the frontend so that we could dynamically remove the entries from it after checking the sale start/end.
Instead of using the "tags" feature, you might want instead to use a custom product field with the corresponding filter and then create the links on the product page yourself to access the products listing with the filter preset for each value selected in the custom field.
That way, the result when you click on the link is a clean HikaShop products listing rather than the results of the tag system of Joomla which isn't really adapted to display products. And on top of that, it will support all the features like published, sale start / end dates, access levels, etc.