No. Two reasons.
THIS adapter will not work. Look close at the molex end of the adapter. There's only two wires - red and black. There's no yellow wire, therefore no 12 volts.
The pinout for a 4 pin CPU fan header is, from pins 1 through 4, CONTROL, SENSE, POWER (12 volts), GROUND. 3 pin header is SENSE, POWER, GROUND. No 5 volts on either.
I think you misunderstood the case manual. Many motherboard have extra fan headers on them to power and control case fans. Or the case fans can be plugged into an IDE drive type power connectopr. I really doubt the case manufacturer recommended powering a hard drive from a motherboard fan connector.
What kind of PSU does not have and IDE drive power connectors? Since IDE drives are still very common, I can't see a "latest design" PSU dispensing with them.