On her 18th visit to the place where she has won six of her 23 majors, Serena Williams took a little over an hour to crush Magda Linette, the 26-year-old Pole who has earned less in her whole career than the American has in just 16 matches this comeback year alone.

If Williams can build on her 6-4, 6-0 win on a sultry night on Arthur Ashe and capitalise on the earlier shock defeat of Simona Halep in her corner of the draw, she has every chance of doubling her season’s earnings of $1.9m.

Bumped up to 17th seed from her world ranking of 26 because of her pedigree and record, Williams looked relaxed, fitter than recently and ready to contend. Her serve was grooved and she hit off the ground with her customary menace. She even threw in a rare drop-shot before serving out the match with her sixth ace. Her movement was better than when she reached the Wimbledon final this year, assured rather than electric.

She said courtside, “It’s just a good feeling to be back out here. The first set was tight, not the easiest. Once I got settled I started to doing what I tried to do in practice. I think I’m getting there. I’ve been training so hard. This momma was a little emotional today.”

On Wednesday, she will play the world No101 Carina Witthoeft who beat the American Carolina Dolehide 6-3, 7-6 (6) on Court No5. If she prevails over the 23-year-old German, she could face her older sister, Venus, in the third round on Friday, when the weather is expected to have cooled a little. Venus survived a stern test in the heat of the day before beating Svetlana Kuznetsova in three sets.

It is three years since the younger Williams sister had her semi-final meltdown here against the Italian doubles specialist Roberta Vinci when striving for a calendar slam, and two years since she again was stopped one match short of the final, by Karolina Pliskova. But there were few visible signs of nerves against world No 68 Linette, who could do little to spoil the opening to the American’s campaign in front of her home fans.

A year ago, Williams, the oldest of seven mothers in the draw at No 36, was only days away from giving birth to Alexis Olympia by emergency C-section, when she survived a pulmonary embolism. There was no guarantee she would even play again, given the serial clotting she endured. “It’s just a bonus to step out here tonight,” she said before going on court.

As Chris Evert observed, “She might be the greatest player of all time, but she’s still human. She’s fallen madly in love with her child, and wants to cuddle her and nurture her, and then she’s got to go out and handle the pressure on the court.”

Halep, the No1 seed at all four majors this year, was the first major casualty of the concluding one – yet it wasn’t as seismic as it might appear, given her capacity to implode in the least likely circumstances. The emotional Romanian, who went out in the first round last year to Maria Sharapova, lasted 76 minutes this time, smashing a racket along the way as her game disintegrated in two stormy sets against the erratic but dangerous Estonian big-hitter Kaia Kanepi, who is 43 places below her in the rankings. A closing forehand, thrashed long with anger and frustration, summed up Halep’s day.

The most remarkable statistics of the match belonged to Kanepi: 28 unforced errors to the Romanian’s nine – and 26 winners to nine again. Like Williams, nobody could accuse her of playing safety-first tennis.