>Note: There is even no record in the log file after mass update.
It's not a problem. Plugin had to run the stored procedure and moved out of stock to the last and return in stock products back to original place. You will not see any information in plugin log.
> However, I think the plugin is not working. It doesn't move the out of stock products to the end of the page.
Take one product (simple product with inventory method 'Inventory method') and set quantity 0. Plugin will change the DisplayOrder value. Check it. Return original Quantity value. Plugin will return DisplayOrder. Check it.