even murdered a fellow paladin.
This didn't actually happen.
He fought Barthilas, essentially disarming him and winding him, but Tirion never delivered the 'death blow'. That honor was given to a horde warrior who essentially threw a spear into Barthilas's back. However, he didn't actually die even then. He survived (or was ressurected) becoming the Magistrate of Stratholme only to die again at the hands of Arthas during the purge.
What it boils down to is this: With the absence of an appointed leader, the highest ranking paladin of the Silver Hand is in command. This is how it works in both older and modern military forces as well. When a senior officer is killed in action, authority over unit command and even in some cases military command is given to the next highest ranking officer until a replacement can be appointed.
Now in the case of the Silver Hand, Tirion is one of the original five paladins. The others being Uther the Lightbringer, Saidan Dathrohan, Turalyon and Gavinrad the Dire. Out of the original five only two remain alive; Tirion and Turalyon. With Turalyon MIA that leaves Tirion the only active member of the original founders, which means that until King Varian as the leader of the Alliance appoints someone to replace him, Tirion is effectively the leader of the Silver Hand.
I have no doubt that when Turalyon does return leadership will pass to him. Not because Tirion is undeserving, he's more than proved himself to the Alliance despite his crimes in the past. But so that Tirion can focus entirely on leading the Argent Crusade. Assuming Turalyon wishes to take leadership at all since as it currently stands Turalyon would have no reason to challenge Tirion for leadership. Hell even Uther at this stage would likely recommend Tirion for leadership were he still alive given all that he's accomplished.