They're running cover for the economic damage that was already done by the COVID unemployment policies, not upcoming damage:

The slowdown in hiring comes just as emergency unemployment benefits that Congress authorized earlier in the pandemic are about to expire. In mid-August, more than 12 million people were receiving some form of jobless aid. Most will receive their final payments next week.

"Companies want to hire more people," said Tim Fiore, who oversees a monthly survey of factory managers for the Institute for Supply Management. "There's no doubt that demand is calling for more people and they can't get them."

Ask anyone in a recruitment role; most companies want more people, but can't find anyone to hire. It's not surprising hiring has slowed right before unemployment benefits expire, because those who are still "collecting" this late in the game won't feel any motivation to find a job until the money stops. I'm sure plenty have been disincentivized from ever getting a job too, and won't reenter the workforce anytime soon.

The shortages and supply-chain disruptions blamed for long product lead times should be evidence enough that demand far outstrips supply (good news for the economy). But who wants to work when the government will give you money for free?