Both the NYT and Washington Post articles on the battle over the succession and the Consumer Financial Protection Bureau (CFPB) neglected to mention the legislative history around the creation of the CFPB. There were competing sections on the order of succession in the event of the director's departure in the House and Senate versions.
One specified that the normal procedure on vacancies, in which the president gets to appoint an acting director, would be followed. The other had language indicating that the deputy director would become acting director until a new director was approved by Congress. This was the language that was used in the final bill. That supports the interpretation of the Democrats that the deputy director should fill in as acting director until Trump nominates a person to be director and that person is approved by Congress.