President Trump may have loved Lindsey Graham, but he loathes the Senate, an institution that Mr. Graham revered for the way it empowered individuals like himself in a way that is currently bedeviling the president and his hard-right allies.

The president’s second-term power grab has gotten scant pushback from Republicans on Capitol Hill, but what little there has been has emanated mainly from the Senate.

Senator John Thune, Republican of South Dakota and the majority leader, has so far refused to acquiesce to presidential demands to gut the filibuster and pass a voting restrictions bill. A handful of other Senate Republicans have bucked the president on policy and nominees, drawing Mr. Trump’s social media wrath and the attendant electoral consequences.

One of Mr. Graham’s multiple self-appointed roles was to interpret the Senate for Mr. Trump — and vice versa. The senator relished his role as an intermediary to the institution that has often angered the president, who has never understood or appreciated its complex ways or why it won’t simply bend to his will as the Republican-led House does.

“It’s overused, but I guess you could call him a Trump whisperer or translator or interlocutor,” Senator John Cornyn, Republican of Texas, said of his South Carolina colleague, whose absence remains acutely felt in the Senate. “We are going to miss him, because the relationship between the president and the Senate is not great anyway, and Lindsey was sort of a bridge that no longer exists.”

The loss of Mr. Graham comes at a particularly fraught time as Mr. Trump has escalated his criticism of the Senate and Mr. Thune. The president is demanding that Mr. Thune keep the chamber in session over the August recess to push ahead with a stalled voter bill and adopt a Republican-only budget framework with funding for the war with Iran and other administration priorities.

Senators, however, are exceedingly reluctant to give up their monthlong recess, particularly in an election year, when there is little to no chance of a legislative breakthrough.

“It’s just a fact, and the facts don’t change,” Mr. Thune told reporters on Monday about the lack of votes for Mr. Trump’s priorities. “Now, if he can change the facts and start moving some of those senators who are noes into yeses, then we’d have a different conversation. But it’s just — that’s just the reality.”

With tensions rising between the Senate and the White House, the White House press secretary, Karoline Leavitt, warned last week that the president’s patience is “running out” on Mr. Thune.

Yet Mr. Trump on Monday downplayed any differences between the two, saying that Mr. Thune has “got to get his job done.”

“It’s fine,” Mr. Trump continued. “I have a good relationship with John Thune. He’s got to get his job done.”

Asked if he had ever considered trying to oust Mr. Thune as majority leader, the president said that “I haven’t even thought of it.”

One reason the president might have said he hadn’t thought of it is that Mr. Thune retains very broad support among Senate Republicans, who show no signs at all of wanting to make a change in their leadership to appease Mr. Trump.

Even some G.O.P. senators unhappy with the legislative agenda Mr. Thune is pursuing remain solidly behind him. All of them see Mr. Thune as a buffer between them and a tempestuous president. Calling for Mr. Thune’s ouster and failing to achieve it would leave the president in a significantly weakened position as far as the Senate is concerned.

After Mr. Graham’s death, the president briefly tried to jump-start his restrictive elections overhaul, which would impose new voting registration and identification requirements, by noting the senator’s support for it and saying its enactment would be a tribute to Mr. Graham. But that effort went nowhere.

His colleagues were fully aware that Mr. Graham had not been particularly outspoken on the voting issue and had instead put his legislative heart and soul into a package of sanctions on Russia. It was that legislation, not the SAVE America Act, that he would have seen as a fitting memorial, and senators were expected later on Tuesday to take their first step toward passing it.

Mr. Graham’s willingness to break with Mr. Trump’s America First tendencies and the increasing isolationist bent of the Republican Party to stand resolute in support of Ukraine won him the admiration and affection of Democrats, despite deep differences on the conduct of the war in Iran.

“Lindsey Graham is the type of senator who is most known for their focus on the role of America in the world,” said Senator Tim Kaine, Democrat of Virginia. “The number of senators who’ve really kind of made it their business to really talk about the role of America in the world, as opposed to domestic policy, has really shrunk.”

Mr. Trump and Mr. Graham didn’t always have a strong relationship. Mr. Graham briefly sought the Republican presidential nomination himself in 2015 and harshly criticized Mr. Trump and his presidential ambitions before they mended fences.

In a phone interview with Fox News on Tuesday morning, the president noted that “we started off a little bit on the rough side” but added that Mr. Graham “really became like a member of the family.”

Mr. Graham, for his part, saw the relationship as partially strategic, wanting to keep a direct line to the president to make his case on issues where he saw Mr. Trump drifting in what the senator considered the wrong direction. He was not afraid to be blunt, and saw the golf course as a convenient venue for candid discussions with Mr. Trump.

“He had a unique relationship with the president, which allowed him to say things to the president that I don’t think a lot of senators would be comfortable saying because he won’t like to hear them,” Mr. Cornyn said, noting that Mr. Trump typically showed little tolerance for the slow-moving Senate. “He’s a businessman and he doesn’t have any patience for process. But that’s what the Senate is all about, is process in order to ultimately get a result.”

Who — if anyone — can now take on the role of Mr. Trump’s Senate confidant remains to be seen, though several Republicans, including Senator John Barrasso of Wyoming, the No. 2 G.O.P. leader, talk frequently with him and carry some influence.

But Mr. Graham’s death has left a void in the Senate’s tense relationship with a mercurial president that will not be easy to fill.