We decided that we will not show the business key in the search table.
There are 2 reasons why we don't want to do this.
1.) We need to query the processInstance rest endpoint for each task of the list (N+1)
2.) we can solve the N+1 problem with custom sql but the workload is higher than the gain of showing the business key in the task search. (You can also see the business key when you click on the process definition link or the process instance link)