I don't know who else could have lead The Order. The Weasleys had their family to worry about, the Hogwarts students are clearly too young. Lupin is unreliable because he becomes a werewolf each month, and both he and Nympadora were concerned with their child.
Most of the wizarding world is opposed to Voldemort, but living in abject fear of him, subservient, and not offering any active resistance (until the battle of Hogwarts). The Order is severely reduced, and hidden from all eyes. It can't exactly put out a job as in The Daily Prophet for a new resistance leader.
That just leaves Moody and Shacklebolt who could be focussed 100% on the work of the Order. So I don't see any reason to doubt this assertion.