Hi MeTaL ,
Apologies for any inconvenience caused.
Product statuses aren't modified until the product is updated and one of the stock-related properties changes. That's why you have to change the field, save the product, and change it again.
I assume you imported the products with the _stock_status field included. You must include the atum_stock_status (ATUM calc stock status in the templates screen) too. That should result in your products being displayed with the correct stock status.

The fastest way to set this field would be by using ATUM Export Pro to update the products. You can create a custom template with only the mandatory fields, plus the atum_stock_status, export the products, modify the CSV with the correct status, and import back using the 'update' option.
Hope this helps,
José Andrés