Yes, that could well work as the ignition starter. In effect turning the Series 2 Ignition turn key starting back to the Series 1 push button starter.
One of the best modifications I have done on my Series 1 is the Ignition Switch Relay to avoid the issue with failing Series 1 Ignition switches.
viewtopic.php?f=9&t=6885
Reading through Heuer’s excellent post above, it describes the reason for the Series 1 Ignition Switch failing to cope with the load on the way to the push button switch and potentially overheating and making the key too hot to touch. Especially true with the later poor quality after market switch contacts. These Series 1 Ignition Switches are switching two circuits, whereas he shows the later Series 2 Switch is only required to switch one circuit. Hence this “Mystery” relay appears to be doing just this.
This is my installation on my Series 1. The double acting relay is switched by the black Earth wire coming from just one set of connections on the original ignition switch - which can see in the centre of the second photo as coming out of the white plastic insulated end piece.
Keep Safe !
Dave