Does that mean every single soldier who is killed in action is automatically the Commander in Chief's fault?
Fault isn't the word i'd use.
But it's his responsibility. If he's got the balls to accept responsibility...
When a ship runs aground or collides with another ship, it doesn't matter if the captain was on the bridge, or touring the engine room or taking a shower. He's responsible for putting his ship in the hands of the people who made the decisions. Even if he left orders 'call me before we get within five miles of shore' and they don't wake him until they they're on the beach, or around the river's bend, or halfway through the coastal highway bridge, he is the one the chain of command will hold responsible.
The only acceptable answer, when the inquest asks 'who was in charge?' is for him to say 'me.'
Trump puts people in place to command the department of defense and execute his policies. He appoints people to each of the branches to execute his policies.
He (or at least his office) issues directives about the priorities to be followed.
In this case he actually had to give the go ahead for the mission. He cannot simultaneously brag about the mission's success and take no responsibility at all for the bad results.
If he wants to sit in the big chair, it's literally his ass on the line.
Now, sometimes men DO die on successful missions.
Investigations try to determine the source of the failure, either in the intel, the planning, the execution or whatever. "Fault" may not lie in Trump's hands.
BUt he cannot disassociate himself by referring to the military as a completely separate entity.