Donald Trump sparked fresh criticism on Thursday by deploying his son-in-law Jared Kushner at a White House coronavirus taskforce briefing and accusing Democrats of launching a fresh “witch-hunt”.

Kushner, a senior adviser to the US president who is married to his daughter Ivanka, made a surprise appearance on the podium and said Trump had instructed him to “break down every barrier needed to make sure the teams can succeed”.

Kushner added: “The president also wanted us to make sure that we think outside the box, make sure we’re finding all the best thinkers in the country, making sure we’re getting all the best ideas.”

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But by way of example, Kushner said Trump became concerned about supply shortages after hearing about them “just this morning” from “friends of his in New York” – implying the president responds to anecdotes rather than the state governor or public health officials.

“We went to the president today,” Kushner continued. “And earlier today, the president called Mayor [Bill] de Blasio to inform him that we are going to send a month of supply to New York public hospital system.”

The vice-president, Mike Pence, later said there would 200,000 masks sent to New York.

Kushner said: “We’ll be doing similar things with all the different public hospitals that are in the hotspot zones and making sure that we’re constantly in communications with the local communities.”

Media reports have suggested that Kushner, a property developer with no medical expertise, is running a “shadow taskforce” – a rival power base that conflicts with the official task force led by Pence.

He said: “I’ve been serving really at the direction of the vice-president. He’s asked me to get involved in different projects. The vice-president and I speak, probably, sometimes five, 10 times a day, but everything that I’m doing is at the direction of the vice-president.”

He glanced over at Pence, who smiled benignly.

Earlier on Thursday, Nancy Pelosi, the House speaker, announced a new House committee would oversee “all aspects” of the federal response to the coronavirus pandemic and did not rule out an investigation in the style of the commission on the 9/11 terrorist attacks.

Such a prospect clearly stung Trump, who compared it to the special counsel Robert Mueller’s Russia investigation and the congressional hearings into his dealings with Ukraine that led to his impeachment.

“This is not the time for politics,” he told reporters. “Endless partisan investigations – here we go again – have already done extraordinary damage to our country in recent years. You see what happens. It’s witch-hunt after witch-hunt after witch-hunt and, in the end, the people doing the witch-hunt have been losing, and they’ve been losing by a lot. It’s not any time for witch-hunts.”

Yet even as the president spoke, the White House was releasing a letter in which he assailed Chuck Schumer, the Senate minority leader from New York. “Thank you for your Democrat public relations letter and incorrect sound bites, which are wrong in every way,” Trump wrote.

Facebook Twitter Pinterest Shortages of equipment have roiled healthcare facilities in New York and across the US. Photograph: Vanessa Carvalho/REX/Shutterstock

“If you spent less time on your ridiculous impeachment hoax, which went haplessly on forever and ended up going nowhere (except increasing my poll numbers) and instead focused on helping the people of New York, then New York would not have been so completely unprepared for the ‘invisible enemy’.”

Along with Kushner, Trump introduced Peter Navarro, the national Defense Production Act (DPA) policy coordinator. Navarro claimed there was a “black market springing up” to drive up prices of protective gear, involving brokers and middle men.

“We are going to crack down on the hoarders,” he said.

Navarro, a hardliner on trade, also used the opportunity claim the pandemic showed “we’re over-dependent on a global supply chain” and was a “vindication” of Trump’s stance on buying American goods and strengthening borders. More than once he praised “Trump time” – shorthand for getting things done fast.

Trump said he was again invoking the DPA to ensure parts were available for the mass production of ventilators. Adm John Polowczyk, in charge of the supply chain, said the federal government had now produced 22.4m pairs of protective gloves, 5.2m face shields and 7,600 ventilators.

But distributing the supplies to the places most in need has been a problem. Trump became defensive under questioning, seeking to shift blame to individual states.

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Some were well prepared, he said, but “in some cases their shelves were bare”. He went on: “By the way, the states should have been building their stockpiles. We’re a backup, we’re not an ordering clerk. Whoever heard of a governor calling up the federal government and saying, ‘Sir, we need a hospital?’”

Trump also faced queries about the economy on a day that saw the number of people filing claims for unemployment benefits surge to a record of more than 6.6 million. He made the astonishing claim: “I will always protect your social security, your Medicare and your Medicaid” – despite having supported cuts in the past.

The president announced that he had taken a second test for coronavirus and, like the first, it came back negative. The second test was much simpler and took about 15 minutes. “I’ve done them both and the second one is much more pleasant,” he said.

America now has more than 236,000 confirmed cases of the virus, according to Johns Hopkins, the highest in the world, and more than 5,600 fatalities. Deborah Birx, the taskforce response coordinator, warned that Americans are not yet taking the risks seriously enough.

“I know you’ve seen the slope in the United States vs the slope in Italy,” she said. “We have to change that slope. We see country after country having done that. I can tell by the curve as it is today that not everyone is following the social distancing guidance. We can bend our curve, but everyone has to take responsibility as Americans.”

Earlier, the treasury secretary, Steven Mnuchin, rejected reports that some Americans might have to wait months for direct payments in the $2.2tn emergency rescue package stimulus package.

“I don’t know where you’re hearing these things,” Mnuchin said. “I told you this would be three weeks; I’m now committing to two weeks. We’re delivering on our commitments to the IRS, which I oversee, within two weeks.”