Yes, ITIL says that any Incident that cannot be resolved (the Service cannot be restored) should be referred to the Problem Management process. The Problem Management Process will do the necessary analysis to determine the Root Cause, CI at fault, and Workarounds. If Problem Management is unable to determine Root Cause, workaround or CI at fault, then it remains an open Problem in the Problem Management database.

The fact that the Problem has never been changed to Known Error doesn’t prevent Problem Management from raising a Request for Change. But the Change Management process must be aware that the risk of failure for a Change applied to an undiagnosed Problem is much higher than for a Change for a Problem that that has been investigated to the Known Error stage. But ultimately it is the responsibility of the Change approvers to either accept the risk and approve the Change Request, or deny it and continue to live with the failed infrastructure.

And it is not unusual to have multiple unresolved/open Problems in the Problem Management Database.

Assuming you have an active PM system and limited resources, the Problem team should focus on high priority Problems. Low priority Problems may never get the required attention for proper investigation.