You can do this with the newly released conditional automations.
It sounds like you have two linked tables: a [Main] table and a [Form] table. The [Main] table has for several different games, and a link to the [Form] table. The [Form] table has a link to a record in the [Main] table and a text field for a person’s name.
I suggest that you add a lookup field in your [Main] table that looks up the person’s name from the linked form.
Then setup your automation to trigger when a record in the [Main] table has a value the link to the [Form]. Use conditions in the automation to determine which column/field in the record is empty. Once the automation finds a blank column/field, update the triggering record to copy the lookup of the person’s name to the proper column, and then reset the linked record field to empty.
You should also setup a filter view in the [Main] table that only shows records where there are available slots/columns/fields. Then have your form only show records in that filtered view.
In my previous message I suggested that you trigger the automation based on the record in the main table that contains the record that you want updated, not from the new record that was created by the form. I also recommend using the “when record meets conditions” as the trigger, instead of when the record enters a view. The condition for the trigger would be based on the backlink to the record created by the form. Then in the conditions for the action, you could see which fields in the triggering record are blank.