How does this soft charge circuit work?
Perhaps there is a hidden center tap on the transformer secondary that connects to the circuit common (L3) through the lower primary lead.

A center-tap connection would answer another mystery about the circuit pointed out in another answer: Because of the series capacitor connected to the collector of the PNP transistor, the timer circuit can't energize RL indefinitely.

With RL initially open (and assuming a center tap), the hypothesis is that the half-wave-rectified DC applied to the contactor is not enough to pull it in. When the timer closes RL after the soft-charge period, both diodes can conduct, and the contactor sees the full-wave-rectified output of the transformer. The full-wave-rectified voltage closes the contactor. Finally, the timer allows RL to open (because of the PNP series capacitor), but the half-wave-rectified voltage is now sufficient to hold the contactor closed.

If this is how the circuit works, I'm curious about the operating margins. Under high-line conditions, the initial half-wave-rectified voltage could risk closing the contactor before it should close. However, doubling the average coil voltage between half-wave and full-wave rectification roughly quadruples the magnetic force, so good margins are plausible with a carefully selected transformer voltage. If margins aren't a problem, an advantage of this kind of approach is that it reduces static power consumption by de-energizing the relay and lowering the coil power of the contactor when the soft-charge cycle is complete.