Democrats “better not screw this up”, a contender for the party’s presidential nomination said on Sunday, as Michael Bloomberg confirmed his late and self-financed entry into the crowded race.

Discussing the impeachment inquiry against Donald Trump, the Minnesota senator Amy Klobuchar told ABC’s This Week: “The American people see this now, they’re listening to it, I’m sure they’re going to be talking about it at Thanksgiving.

“But they want a check on this president, they want an economic check … they also want a patriotism check and a value check. You look at those voters in Kentucky and Virginia [in recent state elections]. They said, ‘Enough is enough,’ they switched the state house and senate in Virginia and they elected a new governor in Kentucky.

“And as Democrats, as we go into this presidential race, we better not screw this up. Because we, right now, have with us a fired-up base, but also independents and moderate Republicans.”

Whether Democrats might be about to screw this up, by either shifting too far left and alienating moderates or by electing a moderate unable to inspire the progressive base, is a subject of constant debate.

On Saturday Monika McDermott, a political science professor at Fordham University in New York, told the Guardian: “Democrats have a real problem here in that they have a bunch of candidates who appeal to different groups for different reasons but they don’t have a clear leader in the group.”

I am not running for chair of the Democratic National Committee. I’m running for president of the United States Amy Klobuchar

Barack Obama has waded into the morass, telling donors in Washington the party has to appeal to “persuadable independents or even moderate Republicans” and saying: “This is still a country that is less revolutionary than it is interested in improvement … the average American doesn’t think that we have to completely tear down the system and remake it.”

Klobuchar is running in the centre of the primary field. She has not polled highly, trailing a top four of Joe Biden, Elizabeth Warren, Bernie Sanders and Pete Buttigieg. The mayor of South Bend, Indiana, another moderate, has surged in Iowa and New Hampshire, the first two states to vote.

Klobuchar told ABC her “constitutional duty” would “come first” if an impeachment trial were staged in the run-up to Iowa in February, obliging her to sit as a juror deciding Trump’s fate.

“But we are moving up in Iowa,” she said. “We are doubling our number of offices. We are adding staff. The same in New Hampshire, South Carolina and Nevada because people are starting to get to know me.

“And they’re starting to understand what I’ve said from the beginning, I am not running for chair of the Democratic National Committee. I’m running for president of the United States. I am someone that always looked people in the eye. I’m blunt. I tell them the truth. I think that’s what we need right now and we need someone who has, yes, bold ideas.”

Klobuchar said those ideas included allowing people with private health insurance to keep it if they wanted to, rather than enforcing Medicare for All. In another potshot at policies championed by Warren and Sanders, she said: “We don’t want to give free college to rich kids.

“Most people agree with me in our party,” she claimed.

Asked what she made of Bloomberg’s entry, which followed that of the former Massachusetts governor Deval Patrick, another moderate professing alarm, she said: “We welcome everyone … people keep coming and going. And I keep steadily going up.”

Bloomberg is set to spend $30m on ads but will not target Iowa, New Hampshire, South Carolina and Nevada, the first four states to vote. He is promising not to take donations, which will disqualify him from the debate stages where Klobuchar has performed well. Nonetheless, the realclearpolitics.com national polling average puts him slightly ahead of Klobuchar, at 2.3% support to 1.5%.

All Democrats need to appeal to certain beliefs among the base. Bloomberg is worth more than $50bn but his first campaign ad presented him as a “middle-class kid who made good”. Klobuchar said she was “the granddaughter of an iron ore miner who worked 1,500ft underground his whole life, the daughter of a union teacher and a newspaper man”.

“But I have actually gotten things done in the gridlock of Washington DC,” she said, “and here’s my difference with the mayor: I have won major rural districts, major suburban districts time and time again and brought people with me. I think that’s going to matter to our voters and that’s the case I will make.

“… Maybe the argument is, ‘Hey, I’ve got more money than the guy in the White House.’ I don’t think they’re going to buy that … They got a guy who’s constantly talking about how much money he has. They’re looking for someone different.”