Do I need both contacts to open/close on a centrifugal switch?
Surgo I understand the goal I repaired a similar motor not long ago 1-1/2 Hp on a 56 motor frame Dayton brand.
The contact set in my unit was 120/240v in the low voltage connection only 1 contact was needed (to tell the truth I did not look to see if it was different for 240v)
These centrifugal switches are used across many sizes and brands.
I got the machine running again by changing over to the unused contact.
I needed a backup or better repair and then found an aftermarket centrifugal switch assembly for under 15$ (I Could not find the correct one on grangers website).
Google or whatever search engine Search for centrifugal motor switch they are sized in mm so you will need your motor shaft size in mm I believe they had 1/2 inch through 1.5”. for example I needed a 16.5 I think it was a L16 it screwed right in.  This was cheaper than $235.00 for a new motor,  35$ Later I had a new motor 2 bearings and the switch.
You may want to check your bearings because end play is what damaged my switch.
I did look for your wiring configuration and came up empty so you may have to trace the connections on the board to the windings they may not be connected as mine were not. They could have a combination of changing the coil configuration + a capacitor and at the time of manufacturing it just used dual contacts where only 1 is needed to bring the cap in and start winding.
If the start winding is held in the motor will overheat because of the extra field coil that provides the torque to start.  so look at the configuration it may be fine but you need to know or you will let the magic smoke out if left on for long.