Washington • In recent days, lawmakers were told that when President Donald Trump ramped up his campaign to pressure Ukraine into helping him against his domestic political rivals, he directed advisers to his personal lawyer. “Talk with Rudy,” he instructed. But one thing lawmakers will not do is talk with Rudy.

Rudy Giuliani was hardly the only offstage character during two weeks of impeachment hearings that ended Thursday. Lawmakers also heard that Mike Pence and Mike Pompeo were in the loop, that Mick Mulvaney organized the political equivalent of a “drug deal” and that John Bolton was adamantly against it.

But among those missing from the House Intelligence Committee’s witness list, besides Giuliani, were Pence, Pompeo, Mulvaney and Bolton. Not that the panel’s Democratic majority was necessarily uninterested in talking with the vice president, secretary of state, acting White House chief of staff or former national security adviser. Democratic leaders have decided not to wage a drawn-out fight to force them to testify over White House objections.

Instead, as the committee wrapped up its public hearings Thursday, House Democrats have opted for expeditious over comprehensive, electing to complete their investigation even without filling in major gaps in the story. It is a calculated gamble that they have enough evidence to impeach Trump on a party-line vote in the House and would risk losing momentum if they took the time to wage a court fight to compel reluctant witnesses to come forward.

But it leaves major questions unresolved. Was Pence told about a suspected link between security aid and investigations of Trump’s political opponents, as one witness testified? Did Pompeo sign off on it? Did Mulvaney facilitate the scheme? Did Bolton ever bring his objections directly to the president? Several current and former officials rushed out statements through aides or lawyers taking issue with testimony about them, but none of them volunteered to offer their own versions of the truth under oath.

Democrats have concluded that in the face of White House refusal to cooperate, it is better to press ahead and simply address the refusal of witnesses like Mulvaney to testify as a plank in a possible article of impeachment alleging obstruction of Congress.

“They should be coming before us,” Speaker Nancy Pelosi said Thursday. “They keep taking it to court, and no, we’re not going to wait until the courts decide. That might be information that’s available to the Senate, in terms of how far we go and when we go. But we can’t wait for that because, again, it’s a technique. It’s obstruction of justice, obstruction of Congress.”

Even some Republican strategists said she had a point. “As a political matter, the longer this goes, it is a real opportunity for Republicans to paint Democrats as unconcerned about the issues voters care more about, and I think Nancy Pelosi is well aware of that,” said Brendan Buck, who was counselor to former Speaker Paul Ryan.

But it leaves some frustrated about the missing pieces. “An impeachable offense should be compelling, overwhelmingly clear and unambiguous,” said Rep. Will Hurd of Texas, one of the few Republicans willing to criticize the president and at one point seen as theoretically open to the possibility of impeachment. “And it’s not something to be rushed or taken lightly. I have not heard evidence proving the president committed bribery or extortion.”

(T.J. Kirkpatrick | The New York Times) Rep. Jim Jordan (R-Ohio), left, and Rep. Mark Meadows (R-N.C.) speak to reporters after a House Intelligence Committee on the impeachment inquiry in Washington on Thursday, Nov. 21, 2019. The House Intelligence Committee, led by its chairman, Rep. Adam Schiff (D-Calif.) is examining the case for impeaching President Donald Trump.

With the White House defying the House, Mulvaney has refused to comply with a subpoena for his testimony while Pence and Pompeo have defied subpoenas for documents but not been asked for testimony. Bolton has declined an invitation to testify and has not been subpoenaed but is awaiting the result of a lawsuit filed by his former deputy, Charles M. Kupperman, asking a judge to decide whether he should listen to the House or the White House.

That case is due for oral arguments in a U.S. District Court in Washington on Dec. 10, but even if the judge rules quickly it could be appealed all the way to the Supreme Court, which would take time.

Another lawsuit seeking to force Donald F. McGahn II, the former White House counsel, to testify in response to an earlier House subpoena in a previous matter may be decided by a judge Monday. But it too could be appealed, and Bolton’s lawyer has suggested that it might not apply to his client since there are separate national security concerns at stake.

None of which would suit the fast-track timetable envisioned by House Democrats. Although more witnesses could still be called, the Intelligence Committee concluded its scheduled public hearings after 12 witnesses and will now focus on drafting a report on the matter. It could also use the coming days to renew its press for the administration to turn over long-sought documents that have become more significant in light of the testimony.

From there, the committee’s report will go to the House Judiciary Committee, which traditionally handles impeachment and will then hold hearings of its own, but generally on constitutional and legal issues rather than fact-finding of its own. After it drafts articles of impeachment, the committee would vote on them and send them to the House floor, where Democrats anticipate a vote by Christmas.

In theory, if witnesses like Bolton do agree to testify or are compelled by a court, they could still be called before the Judiciary Committee. And for that matter, if the House does impeach Trump and sends the case to the Senate for a trial that would open sometime after the new year, additional witnesses could still be called then, too.

But the two weeks of public hearings showed how much remains fluid. Gordon D. Sondland, the ambassador to the European Union and a key figure in the pressure campaign, amended his original closed-door testimony after other witnesses contradicted him. Others like William B. Taylor Jr., the acting ambassador to Ukraine; Kurt D. Volker, the special envoy for Ukraine; and Laura K. Cooper, a Defense Department official, offered new information after their original interviews when reminded by their staff or other witnesses.

Some of those who have not testified are aggrieved at their portrayals over the last two weeks. Mulvaney protested testimony Thursday by Fiona Hill, a former Bolton deputy, that put him at the center of the pressure campaign.

“Fiona Hill’s testimony is riddled with speculation and guesses about any role that Mr. Mulvaney played with anything related to Ukraine,” his lawyer, Robert N. Driscoll, said in a statement. But the statement did not explain what role he did play, leaving the committee to guess.

In Mulvaney’s case, he has made statements that Democrats, at least, will consider evidence even if it was not under oath. During a briefing for reporters last month, Mulvaney admitted that Trump suspended $391 million in U.S. security aid to Ukraine in part to force Ukraine to investigate a debunked conspiracy theory involving Democrats during the 2016 presidential campaign. Mulvaney later tried to take it back.

Those comments as well as statements by the offices of officials like Pence and Energy Secretary Rick Perry raise the question of whether they have effectively waived any claim of immunity from testifying because they have publicly addressed the matter, according to lawyers. But Democrats may not take the time to litigate the question.

Rep. Tom Malinowski, D-N.J., who serves on the Foreign Affairs Committee that was involved in the Ukraine investigation at an earlier stage, pointed to Mulvaney’s public acknowledgment about the link between aid and an investigation as well as other testimony about figures like Bolton and Pompeo.

“I very much want to hear from them,” he said. “But if they lack the courage of their colleagues to testify under oath, we can assume that what we’ve learned about their views and actions is true.”

An impeachment proceeding is not the same as a criminal court process, of course, and the standard of evidence is not the same. The House can move forward with whatever evidence a majority considers sufficient. And to the extent that the processes can be compared, an impeachment would be the political equivalent of an indictment, signaling that there is enough evidence to merit a trial in the Senate, though not necessarily enough to convict.

Still, even in the relatively quick investigation conducted in the two months since Pelosi formally opened the impeachment inquiry, the basic facts of what happened have been established and to a greater or lesser degree verified by different witnesses.

“The reality is there’s not much ambiguity about what took place here,” Buck said. “We know what happened, and now members and voters have to decide whether it rises to the level of removing him.”