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You can measure the resistance of the motor coils to discount theose being burnt out. It takes a few seconds for a multimeter to stabilise at a reading

The 3 pins Common, Start and Run.
S-R (6K ohm? ) highest reading through both coils.
C-S (4K) resistance of Starting windings
C-R (2K) resistance of Run windings
Any pin to Earth should be open.

They're reading much lower - 22/45Ohms. I Guess this means the motor's screwed?

Does the capacitor (silver cylinder to the left of your motor) seem to be bulged or leaking?

No

Do you have a multimeter to test capacitance?

Yes, seems to be ok

You can test the coil of the relay for continuity and across the switched terminals but it can be difficult to tell if it's actually switching when in situe.
Does it smell burnt or have a discoloured area?

Edit.
Some current will flow through the run winding, even if the motor is stationary. Hence getting warm.
I couldn't hear it, but does yours actually run for a few seconds and then trip off, or just sit there humming before tripping.

The later I think, there was a slight hum between it trying to kick in and the 2nd click (overload?) but nothing that sounded like the motor running...

Edit: no signs of burning or arcing on the relay either