How could Gordon Ramsay close restaurants in Kitchen Nightmares

1. It's a reality show.
   the showrunners have the purse strings & the restaurant owner's full co-operation [whaterver fireworks may be going off, they had to sign something that gives the show the ultimate say]
   The owner is also hoping for a decorative do-over, worth thousands, as 'reward' for their co-operation in the making of the show.

2. See 1.

A late 3. is that some of the violations he discovers would land the owners in court with health & safety issues, hefty fines & potential permanent closure. He is actually doing them a favour by yelling "Shut it down!" no matter how over-dramatically for the cameras.

Scores on the Doors wasn't invented back then. Who knows how some of those establishments managed to hide their practices from the health inspectors.

I've actually eaten in one of the restaurants Gordon improved - ostensibly very successfully - which he then revisited a couple of years later to find they'd moved to a new far larger establishment, based on the popularity generated by the first show. Unfortunately they had fallen back into their old habits of pre-cooking everything & flash-frying it for service; not a good way to do it. He tried again to persuade them to change. They did, for a while, but old habits die hard.

He was right, the food was barely edible. They have since closed for good.