On a day of alarming polling about attitudes to political violence and fears for US democracy, and as the first anniversary of the Capitol attack approached, a Republican member of the House committee investigating the events of 6 January 2021 had a stark warning for her party.

“Our party has to choose,” Liz Cheney told CBS’s Face the Nation. “We can either be loyal to Donald Trump or we can be loyal to the constitution, but we cannot be both.”

Trump supporters attacked Congress in an attempt to stop certification of his defeat by Joe Biden, which Trump maintains without evidence was the result of electoral fraud. Five people died around a riot in which a mob roamed the Capitol, searching for lawmakers to capture and possibly kill.

On Sunday, Cheney and Bennie Thompson of Mississippi, the committee chairman, again discussed the possibility of a criminal referral for Trump over his failure to attempt to stop the riot or for his obstruction of the investigation.

Speaking to ABC’s This Week, Cheney said there were “potential criminal statutes at issue here, but I think that there’s absolutely no question that it was a dereliction of duty. And I think one of the things the committee needs to look at is … a legislative purpose, is whether we need enhanced penalties for that kind of dereliction of duty.”

Thompson said subpoenas could be served on Republicans in Congress who refuse to comply with information requests of the kind which have led to a charge of criminal contempt of Congress for Steve Bannon, Trump’s former strategist, and a recommendation of such a charge for Mark Meadows, Trump’s chief of staff.

The Democrat told NBC’s Meet the Press the committee was examining whether it could issue subpoenas to members of Congress, immediately Jim Jordan of Ohio and Scott Perry of Pennsylvania.

“I think there are some questions of whether we have the authority to do it,” Thompson said. “If the authorities are there, there’ll be no reluctance on our part.”

Last month, the committee asked Jordan for testimony about conversations with Trump on 6 January. Jordan told Fox News he had “real concerns” about the credibility of the panel.

Perry was asked for testimony about attempts to replace Jeffrey Rosen, acting head of the justice department, with Jeffrey Clark, an official who tried to help overturn Trump’s defeat.

Perry called the committee “illegitimate, and not duly constituted”. A court has ruled that the panel is legitimate and entitled to see White House records Trump is trying to shield, an argument that has reached the supreme court.

Sunday saw a rash of polls marking the anniversary of 6 January.

CBS found that 68% of Americans saw the Capitol attack as a sign of increasing political violence, and that 66% thought democracy itself was threatened.

When respondents were asked if violence would be justifiable to achieve various political ends, the poll returned an average of around 30%. A survey by the Washington Post and the University of Maryland said more than a third of Americans said violence against the government could be justified.

ABC News and Ipsos found that 52% of Republicans said the Capitol rioters were trying to protect democracy.

Other polling has shown clear majorities among Republicans in believing Trump’s lie about electoral fraud and distrust of federal elections.

On CNN’s State of the Union, Larry Hogan, Maryland governor and a moderate Republican with an eye on the presidential nomination, said: “Frankly, it’s crazy that that many people believe things that simply aren’t true.

“There’s been an amazing amount of disinformation that’s been spread over the past year. And many people are consuming that disinformation and believing it as if it’s fact. To think the violent protesters who attacked the Capitol, our seat of democracy, on 6 January was just tourists looking at statues? It’s insane that anyone could watch that on television and believe that’s what happened.”

Cheney told CBS the blame lay squarely with her own party.

“Far too many Republicans are trying to enable the former president, embrace the former president or look the other way and hope that the former president goes away, or trying to obstruct the activities of this committee, but we won’t be deterred. At the end of the day, the facts matter, the truth matters.”

Her host, Margaret Brennan, pointed out that Republicans across the US, some in states where Trump’s attempt to steal the election was repulsed, are changing election laws to their advantage.

“We’ve got to be grounded on the rule of law,” Cheney said. “We’ve got to be grounded on fidelity of the constitution … So I think for people all across the country, they need to recognise how important their vote is for their voices. They’ve got to elect serious people who are going to defend the constitution, not simply do the bidding of Donald Trump.”

Cheney faces a primary challenger doing Trump’s bidding and enjoying his backing. The other Republican on the 6 January committee, Adam Kinzinger of Illinois, will retire in November rather than fight such a battle of his own.

Cheney said she was “confident people of Wyoming will not choose loyalty to one man as dangerous as Donald Trump”, and that she will secure re-election.

She also notably did not say no when she was asked if she would run against Trump if he sought the nomination next time.

On ABC, Cheney was asked if she agreed with Hillary Clinton, who has said a second Trump presidency could end US democracy.

“I do,” Cheney said. “I think it is critically important, given everything we know about the lines that he was willing to cross.

“… We entrust the survival of our republic into the hands of the chief executive, and when a president refuses to tell the mob to stop, when he refuses to defend any of the co-ordinate branches of government, he cannot be trusted.”