Product not re-enabled after order cancellation?

If there qty of the item was 1 and the item was ordered, then the item is disabled as it has quantity 0. However if the order is deleted from admin with "restock product quantity" option, the quantity is restored back to 1 but the product stays disabled and does not show up in catalog.

That is correct, the return to stock does not assume a return to products_status 1

You will need to do that ...

Note: you can have products_status remain at 1 even when products_quantity is <= 0 and the Product will change to Sold Out and not be allowed to be added to the cart ...

Then if you do return the product to stock, the product will automatically also be available as the Product will no longer be products_quantity <= 0 and the products_status will already be 1 ...

Well, that's not exactly what I wanted - I wanted a product status to be automatically Disabled when its quantity is <=0 and Enabled when its qty >=0.
However, this far it would be better to make a product stay visible (not change its status to disabled) when it's qty falls to 0. But how can I do this?

The settings for the Stock are in the Configuration ... Stock ...