That service socket puts 12V to the starter relay (2/35), the other side of the relay coil is grounded through the PNP switch on the transmission. If your car cranks over using the ignition key, it means the system (Ign sw. PNP sw. starter relay, starter solenoid) is OK. The starter relay is the one closest to the driver's side fender in the back row of relays in the underhood fusebox. Does it click when you try the service socket? Put a finger on it and you should feel its internal contacts activate.

You only need a piece of wire about 10cm (4in) long to test the service socket.
The fuse (#18) closest to the engine in the rear row of fuses in the underhood box has 12v at all times. Touching a wire to the bit of metal exposed at the fuse top and to the service socket will crank the engine if all is well in the starter circuit. I've tried this, it works. The current flow is just 1/6th amp - enough to trigger the starter relay.
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Bob: Son's XC70, daughter's 940, my 81 and 83 240's, 89 745 (V8) and S90. Also '77 MGB and some old motorcycles