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In this story, the main character enters his autonomous car (no steering wheel or pedals), which takes him out of the city to a picnic ground. After a pleasant afternoon, the car takes him back to the city, but travels through a tunnel. The tunnel entrances close. When the entrances open the cars exit, empty, clean, and slightly damp. This is the means of population control - car occupants have been liquefied and flushed away.

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This sounds like "The Tunnel Ahead", a 1961 short story by Alice Glaser, also the answer to the question Looking for story about population control via beach vacation; first published in The Magazine of Fantasy and Science Fiction, November 1961, available at the Internet Archive.

From Alice Glaser's page at The Encyclopedia of Science Fiction:

(1929-1970) US writer and editor, a Radcliffe graduate and social activist who lived in Paris for several years before returning to New York to become an associate editor of Esquire magazine (see Slicks). Her one sf story is the frequently anthologized "The Tunnel Ahead" (November 1961 F&SF), reminiscent in some ways of Shirley Jackson's "The Lottery" (26 June 1948 The New Yorker). In a overcrowded, Dystopian future when the US population has reached one billion, a family returning to Manhattan (see New York) from the crowded beaches must drive through an 8500 foot tunnel – the only way into or out of the city. At random intervals some ten times per week, the tunnel is sealed off and the occupants of the 700-odd trapped vehicles are gassed to help reduce Overpopulation; nevertheless this risk offers city-dwellers their only excitement in life. Glaser also contributed nonfiction to Esquire, but published no other fiction. A short film adaptation of "The Tunnel Ahead" directed by André Øvredal, who made Trollhunter (2010), is in post-production in Norway.

As for the plot points you mentioned:

In this story, the main character enters his autonomous car (no steering wheel or pedals),

The car has both manual and automatic modes, so it does have steering wheel and pedals:

Tom's muscles ached with their familiar cramp. He ran his hands uselessly around the steering wheel, wishing he had something to do, or that there were room to stretch in the tiny car, then felt instantly ashamed of his antisocial wish. Naturally there was nothing for him to do because the drive, as on all highways, was set at "Automatic". That was the law.

which takes him out of the city to a picnic ground.

An outing at the beach:

Besides, it hadn't been a bad day, all things considered. Five hours to cover the forty miles out to the beach, then of course a couple of hours waiting in line at the beach for their turn in the water. The trip home was taking longer: it always did.

They had their "picnic lunch" in the car:

The other three hadn't spoken for hours which, of course, was as it should be. Jeannie had fed them a purposely heavy lunch in the car, steakopop and a hot, steaming bowl of rehydrated algaesoup from the thermos, and they had each had an extra dose of tranquillizers for the trip.

After a pleasant afternoon, the car takes him back to the city, but travels through a tunnel.

They had reached the Tunnel entrance. Jeannie was silent. She glanced at her watch, irrationally. Tom pressed the tranquillizer button and the drawer shot out, but she shook her head.

"I hate this, Tom. I think it's an absolutely lousy idea."

Her voice sounded almost savage, for Jeannie, and Tom felt a little shocked.

"It's the fairest thing", he argued. "You know it perfectly well."

Jeannie's mouth had been set in a stubborn line. "I don't care. There must be another way."

"This is the only fair way", Tom said again. "We take our chances along with everybody else."

The tunnel entrances close.

The viewpoint characters get through safely this time:

Tom's hand drummed at the wheel. The maverick ahead had edged back into line. They started movement again. They picked up speed. They were out of the Tunnel.

Jeannie picked up her knitting and shook it, sharply. Then she dropped it as though it had bitten her fingers. A bell was clanging over their heads, not too loud, but clear. Just behind their bumper a gate swung smoothly into place.

Jeannie turned to look back at the space behind them where the Italian family in the bright blue car, and others, had been. There were no cars there now. She turned back, to stare whitely through the windshield.

When the entrances open the cars exit, empty, clean, and slightly damp. This is the means of population control - car occupants have been liquefied and flushed away.

The occupants are gassed and their bodies removed, nothing said about liquefaction:

Tim was figuring. Two minutes for the ceiling sprays to work. Then the seven hundred cars in the Tunnel would be hauled out and emptied. Ten minutes for that, say. He wondered how long it was supposed to take for the giant fans to blow the cyanide gas away.

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