A/C compressor will not engage

About a week ago the A/C stopped working. All of a sudden just went from blowing forty degree air to nothing over the course of one minute.

The system does have refrigerant and is holding pressure.

Replaced the A/C pressure switch so that is not the issue.

Checked the relay (A/C clutch relay), getting power to the relay and the relay itself is good.

Now, here is the twist.

When I manually put twelve volts to the compressor and jump it, then it kicks on, seemingly runs fine, blows cold air. Works perfectly. It just will not engage itself. There is no power going to the compressor.

What is going on here? Is there another relay, or sensor, or fuse or something I do not know about? Something that is preventing power from making it to the compressor? Is it a wiring issue somewhere?

I have been told by a couple people that the compressor itself (actually they said the clutch) is bad and needs to be replaced, but that does not seem right. I mean if there is no power going to the compressor through the connector plug what good is putting a new one in going to do. Connecting a dead plug to a new compressor is not magically going to make it have power. Right?

Any advice or ideas would be much appreciated. Thanks in advance for the help!