Kamala Harris has refused to be drawn into a war of words with Joe Manchin over the West Virginia senator’s attempt to sink the Build Back Better spending plan, saying: “The stakes are too high for this to be in any way about any specific individual.”

The vice-president, whose vote in the 50-50 Senate would have passed Build Back Better had Manchin (and the other 49 Democrats and independents) stayed onboard, was speaking to CBS News.

But Manchin issued a dramatic “no” in an interview with Fox News Sunday, enraging the progressive wing of the Democratic party as well as the White House, which issued a stinging rebuke.

“I don’t have any personal feelings about this,” Harris insisted. “This is about let’s get the job done. Let’s get it done.

“I refuse to get caught up in the what might be personal politics. The people who are waking up at three o’clock in the morning worried about how they’re going to get by, they could care less about the politics of DC.”

Build Back Better, valued at around $1.75tn, aims to boost social and health care as well as target spending at the climate crisis and other Democratic priorities.

Harris said: “Let’s talk with families who say I can’t afford to do the basic things that I need to do as a responsible adult, like care for my children, care for my older parents, or afford to get life saving medication like insulin.”

Asked how the Democrats could do that without Manchin – as many in the party have said they will try to do, if they are not able to turn him round – Harris said: “You don’t give up? That’s how we do it.”

Manchin does appear to have personal feelings on the issue, having told a West Virginia radio station on Monday he reached “wit’s end” before deciding to drop his bombshell on Fox.

“This is not the president, this is staff … they drove some things that are absolutely inexcusable,” he said. “I just got to the wit’s end of what happened.”

Manchin also said he had been “far apart, philosophically” with Democratic leaders for months.

“We’re in a 50-50 Senate, you all are approaching legislation [as if] there is 55 or 60 Democrats,” he said.

Some fear Manchin could switch allegiance from Democratic to independent or even Republican – as every other official in major office in his state has done. Such a move would jeopardise or end Democratic control of the Senate.

On Monday, Manchin said he “would like to hope there was still Democrats that feel like I do”, but said that could change.

The Senate minority leader, Mitch McConnell, said he would “certainly welcome” Manchin into Republican ranks.

“He doesn’t fit well over there,” McConnell said, “but that is a decision ultimately that he has to make. We certainly welcome him to join us if he was so inclined.”

McConnell also told the Guy Benson Show podcast he was “shocked at the vitriol” in the White House rebuke of Manchin.

“And basically it seemed to me that they were calling Senator Manchin a liar. I think that was not smart. This is a 50-50 Senate. It’s going to be 50-50 for another year, and believe me, this is not how I would handle a disappointing vote like that.”

Joe Biden’s press secretary, Jen Psaki, told reporters in Washington she would not “relitigate” Manchin’s announcement on Sunday and the White House rebuke, which was issued in her name.

West Virginia is a major coalmining state. Also on Monday, a prominent coal union urged Manchin to reconsider his opposition to Build Back Better, not least because it would extend benefits, due to expire at the end of the year, to miners suffering from black lung disease, and encourage investment in jobs for former miners.

“For those and other reasons, we are disappointed that the bill will not pass,” said Cecil Roberts, president of the United Mine Workers of America.

“We urge Senator Manchin to revisit his opposition to this legislation and work with his colleagues to pass something that will help keep coal miners working, and have a meaningful impact on our members, their families and their communities.”