I have something like the product you’ve found. Except mine has a single USB-C plug. My device can’t power 3.5"HDDs nor optical drives. But it will power 2.5"HDDs and of course SSDs.

Your device has two USB plugs, one blue and another white. That the white plug is auxiliary power seems plausible. The USB standard specifies 0.5A max current per port. So two ports, 2.5W each, is 5W. 12V and 5V both 0.75A is 12.75W. So unless your computer has charging ports, capable of delivering extra current, it won’t work.

The compact device is nice, when you’re not working with 3.5" drives, but I have found the universal solution to be something with a separate SATA power connector, like this reichelt.com/…/usb_2_0_ide_sata_adapter_with_back…

It’s bulky, but it works.