If the pump engages when you manually close the relay, then the relays coil side is not getting power to pull the relay IN..that is why I had you check the fuse # 7...that fuse does not feed the pump clutch, it feeds the signal to the relays PRIMARY COIL. That is where you have to check.
So, IS the coil contacts at the relay getting voltage from fuse 7 ??? ..or is the coil in the relay OPEN ???
Those are the questions you have to be addressing. The test for those 2 questions is an ohm meter reading across the relay coil [ 85/86] for coil verification and a voltage reading at those terminals to energize the coil.

No coil reading equals NO PUMP
No Voltage at coil terminals equals NO PUMP.
Bad or poor connection at fuse equals NO PUMP.

By closing the relay manually, you have eliminated the LOAD side of the circuit and verified that circuit/ part to be OK..so we know you have a relay coil feed/sensor side problem. [ unless the pump IS coming ON...that is why I asked " IS the pump coming ON ???"]

There are criteria to get coil feed , but we don't want to go there if you do see the pump ever run. If the pump runs, then the problem is in the valves and piping.
So, answer my questions and I can then give you the next question to help track it down.
You seem to be confused about the pump...it only RUNS when the clutch is energized , so when I ask is pump RUNNING, I don't mean anything else except "Is the PUMP CLUTCH being pulled IN...
I think you have that understanding, so before getting a relay, pull it and confirm the coil windings w/ohm check....you may have never seen or read of a relay to be the problem, but I have seen many relays with OPEN coils...so we go by coil ohm readings , not by never seen one do that before............
If you jumper the relay pins for clutch , then the AIR system plumbing and Switch over Valves can be checked by monitoring the 02 sensor...but check the relays coil first... . and also recheck that fuse ..they will fool you and look fine..I use a test lamp on both sides to be sure.

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A Dalton