I solved the error the hard way by first deleting the index .opendistro-ism-config and then reconfigure the SM policy.
/!\ This action removes every ISM you could have configured until then on your OpenSearch cluster (every action triggered by the JobScheduler).
My intuition is that the JobScheduler couldn’t validate the configurations stored in the index .opendistro-ism-config. I guess I could have cleaned it manually instead of deleting everything. The issue is clearly linked to the presence of a corrupted document within the index that leads to JobScheduler error.
On the other hand, I still don’t understand what caused the index to be corrupted.
I had the exact same issue on my preprod cluster and found the root cause of the issue.
TLDR; The user used to create Snapshot Management Policy did not exist anymore leading to the issue.
The error was indeed caused by a corrupted document in the .opendistro-ism-config index, more specifically by the document describing the Snapshot Management Policy that I configured previously, say MY_SMP.