“It’s tremendous news that the U.S. has ended Baghdadi’s bloody jihad,” added Sen. Ben Sasse, a Nebraska Republican and member of the Intelligence Committee. “The President made the right call to take out this bloodthirsty monster who led ISIS as it raped and pillaged its way through Iraq and Syria.”

But Sasse and others also warned in a statement about letting up the pressure. “As Americans celebrate this victory, we must remain clear-eyed that this is no time to let off the gas: Baghdadi is gone but another animal will take his place as ISIS works to regroup.”

“Removing the leadership of terrorist groups is not on its own a decisive win. It never has been,” said Eric Robinson, a former intelligence official. “Saying that the caliphate is going to crumple as a result of this is just wrong. It will endure.”

The death of its leader won’t mark the defeat of ISIS, agreed Michael Nagata, a retired Army lieutenant general and former senior intelligence official who fought the Islamic State’s predecessor organization in Iraq and was the top special operations commander in the Middle East during the early years of the ISIS campaign.

“I’ve never seen the death of a senior leader be the catalyst for the elimination or destruction of a powerful, well-entrenched, global terrorist movement. It’s a necessary step but it’s never a decisive step,” Nagata said in an interview.

“There are a lot of parallels to be drawn to the impact of Osama bin Laden’s death,” said Nagata, who was also the senior U.S. military official in Pakistan at the time of the 2011 raid that killed the founder of al-Qaida. “It was important at the time and had enormous symbolic value but it is mostly strategically irrelevant now. That’s the trajectory I expect the impact of Baghdadi’s death to follow.”

Retired Army Lt. Gen. Mick Bednarek, who was the top U.S. officer at the U.S. embassy in Baghdad during ISIS' rise in 2014 and 2015, said in an interview it is “incredible news and speaks highly once again of our collective intelligence agencies and more specifically our special operators.”

But Bednarek, who four years ago called the battle against ISIS “the fight of our lifetime,” cautioned against taking too many victory laps.

“When you eliminate the head or figurehead, does that mean the demise of the Islamic State terror organization? Absolutely not. Who is the next emergent leader? It is going to have an impact, but it is not going to be the be all end all where we can rest on our laurels."

Trump used unusually vivid, even gory, language in describing Baghdadi's final moments — descriptions that that some regional experts feared could further inflame extremists in the region.

"He died after running into a dead-end tunnel, whimpering and crying and screaming," Trump said. "The compound had been cleared by this time, with people either surrendering or being shot and killed. Eleven young children were moved out of the house un-injured. The only ones remaining were Baghdadi in the tunnel, who had dragged three children with him to certain death. He reached the end of the tunnel, as our dogs chased him down. He ignited his vest, killing himself and the three children. His body was mutilated by the blast, but test results gave certain and positive identification."