To answer the question regarding updating multiple linked records with an automation, it's possible but not the way your automation is set up. It could go as follows:
Trigger: If a record is updated (the field with multiple records)
1. If the conditions you want are met.
2. Update record into the new field as desired. The record id. would be the airtable trigger and select the field you want and put the linked record name.
However the above described is not the current automation you have. To fix your current automation, the record id. in the screenshot you provided should not be the linked record. It should either be the trigger record id. or you need to create a find record step beforehand to make sure you are getting the right record id. number.
Let me know if this fixes the issue!
Thank you Valentino. I found a solution with the "Repeating Groups" https://support.airtable.com/v1/docs/repeating-groups-of-automation-actions?_gl=1*fr096b*_ga*Njc1NTA...
Now it first find the multiple linked records and after run the update on all of them.