Why is my 3.5" SATA hard drive not spinning up via USB?

It does matter what size the drive is. Large 3.5" hard drives require more power then what even two USB ports can provide. For reference, a single USB port can provide a maximum 500 mA of current. A Western Digital WD3200AAJS, on the other hand, requires 1444 mA at idle, and 1608 mA when reading/writing files to/from the drive.

It is part of the USB specification that if a device attempts to draw too much power, it is simply disconnected from the target system - this is why, while your adapter itself shows up, you can't access the drive. There is no jumper setting to change, SATA drives don't have any.

What you're trying to do is, quite bluntly, impossible. You'd need the power of almost four USB ports to get the HDD working, which is also why you never see full 3.5" drive enclosures without an external power supply.

Your only option is to power the drive from the computer, power it using an external power supply, or if you can find any (reliable) +5V and +12V DC source, you could hack together your own. Or you could just buy a USB HDD enclosure that has the power source.

A 3.5" drive requires both 5v and 12v, wheras 2.5" drives only need 5v. USB only supplies 5v. USB adapters/enclosures for 3.5" drives have a separate 12v input in addition to the 5v, which can be supllied by the USB bus or by the power supply, in which case the PSU is supplying both voltages.

A single USB cable is not sufficient to power most 3.5" drives (which is why you might have seen some crazy USB Y-adapters on certain enclosures-- those are to draw power from two ports and power the drive without an external adapter), but it can power a 2.5" drive. You will need to get a proper external enclosure that can support the power requirements of a 3.5" drive.