Javiera Martinez, her partner Daniel Capurro and their three children aged 14, seven and eight were supposed to be flying to Chile on Friday to visit relatives they had not seen in three years.
But after their 8am Qantas flight from Melbourne was delayed by half an hour, baggage handling and airport transfer delays in Sydney meant they couldn't make their 11.30am LATAM Airlines flight to Santiago.
Ms Martinez said the airline's procedures at the airport were chaotic.
"We think Qantas didn't behave appropriately, I got berated by the person at the counter, they never apologised, they never assumed any responsibility at all," she told AAP.
"It was a rude conversation, we have been mistreated badly I would say."
The PCR tests they need to travel have now expired and they will have to re-take them as they wait for seats on the next flight to Santiago departing Sunday.
The airline has apologised and paid for a night's accommodation in Sydney.
"We sincerely apologise that the family missed their connecting flight on another airline due to delays moving through Sydney airport on Friday," a Qantas spokesperson told AAP.
The family is among many affected by travel hold-ups amid the busiest travel period in two years, with Sydney, Melbourne and Brisbane airports warning passengers to arrive two hours before domestic flights.
On Friday, Qantas CEO Alan Joyce clarified earlier comments that passengers were "not match fit", saying the delays were not their fault.
The airline shed thousands of staff during the pandemic, and outsourced ground crews in a decision that was challenged in court.
On Friday, about 78,000 domestic passengers went through Melbourne Airport, with construction work at the terminals adding to congestion and queues.
In Sydney, domestic passenger numbers have nearly doubled in the past six months.
Sydney Airport General Manager of Operations Greg Hay warned travellers to be ready to go through security when they arrive at the terminal.
"We've also noticed people are a little out of practice with security protocols, for example not taking laptops and aerosols out of their bag at screening points, which is contributing to the queues," he said in a statement.
Amid the surge in passenger numbers, the company providing security services at the airport, Certis Security Australia, is also attempting to rebuild its workforce, Mr Hay said.
More than 51,800 travellers went through Brisbane airport on Friday, making it the busiest day in more than two years.
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