Why was the Austro-Hungarian government not really concerned whether Serbia itself was involved in Archduke Ferdinand's assassination?
The reason for this is that the government of Austria-Hungary really wanted a war with Serbia. They did not really care what the truth was. Instead, they wanted a way to force Serbia to go to war with them.
Austria-Hungary wanted Serbia destroyed or seriously hurt for two major reasons. First, the presence of Serbia was serving to inflame pan-Slavic nationalism among Austria-Hungary's Slavic subjects. Serbia was helping to fan these flames and was even thinking about trying to take Bosnia (a Slavic region of the empire) away from Austria-Hungary. Secondly, Austria-Hungary wanted more territory in the Balkans and it specifically wanted to prevent Serbia from gaining a port on the Adriatic. Austria-Hungary had become more and more worried about both of these issues in the wake of the Second Balkan War.
For these reasons, Austria-Hungary was eager to go to war with Serbia. That meant that they would not really care about the truth and would take any reason they could find to start a war.