The plan seemed mutually beneficial: President Trump would bask in an adulatory slice of Americana, and the Boy Scouts of America would host yet another sitting president at its national jamboree.

But Mr. Trump’s decision to mix a barrage of political remarks into a speech that has traditionally been an uncontroversial stream of upbeat oratory has enraged many parents and former Scouts, thrust the Scouts once again into the middle of the nation’s culture wars and provided yet another example of the unusual and polarizing nature of the Trump presidency.

The Scouts, plainly sensing a new threat that supporters feared could undermine a movement whose membership is already sagging, said in a statement that the group was “wholly nonpartisan and does not promote any one political position, candidate or philosophy.” The group added that its traditional speaking invitation to the president was “in no way an endorsement of any person, party or policies.”

The Greater New York Councils of the Boy Scouts was somewhat blunter, saying Scouting is an apolitical organization, and “it is inappropriate for any President to use the Jamboree as a backdrop for political statements.’’

It was far from clear that Tuesday’s efforts by the Scouts would calm an uproar that began even before Mr. Trump concluded his address on Monday night. Although Scouting offices and social media accounts were besieged with messages condemning the president’s appearance, others celebrated Mr. Trump’s speech in West Virginia. “Trump gave a great speech to the Boy Scouts and they chanted back “We love Trump!,’’ read a Twitter post in the name of Shaun Hough (“Philosopher, conservative, libertarian”). “I love it!!”

Either way, the firestorm was unwelcome, and unexpected, during one of Scouting’s most important events, a gathering that attracts people from around the world and, very often, presidents, who have typically spoken about service, values and citizenship, not partisan politics.

Reaction to the speech, delivered to an enthusiastic audience of saluting and cheering Scouts, was immediate and visceral.

Glenn Elvig, a Minnesota artist who fondly recalled receiving a letter from Richard M. Nixon congratulating him on achieving the Eagle rank decades ago, said he called a Scouting office for hours on Tuesday to express his dismay. He kept getting a busy signal.

“I would like a public denouncement of what happened yesterday and reaffirmation of the values I think I learned in Scouts,” Mr. Elvig said. “If they can’t do that, I will be returning my medal.”

Brian Alexander, who earned the Eagle rank as a teenager in Ohio, said he was also outraged after seeing clips of Mr. Trump’s speech. Mr. Alexander sent several messages to Boy Scout officials on Twitter, calling the address “a disgrace.”

“Based on my experience with scouting, the point is you’re supposed to grow up to be someone not like Donald Trump,” Mr. Alexander, 32, said in an interview. “You’re supposed to grow up to be someone like John McCain or Barack Obama.”

Mr. Alexander’s complaint, echoed by many other veterans of Scouting, was not that Mr. Trump had been invited to the jamboree, but that he had opted to use his appearance as he did.

“I can guarantee there were people in that audience of 40,000 who agreed with what he said,” said Alvin Townley, the author of “Legacy of Honor,” a frequent gift to newly minted Eagle Scouts. “There were also plenty of people in that audience who did not agree with what he said, and there were probably plenty of people, whether they agreed with him or not, who felt it was an inappropriate venue to discuss politics.”

In the opening moments of his appearance, it seemed that Mr. Trump, who was not a Boy Scout as a youth, would mostly avoid talking about the partisan clashes that have riven Washington.

But Mr. Trump, the 19th occupant of the White House to also serve as the honorary president of the Boy Scouts of America, ultimately speckled his remarks with political speech that proved startling at a Scout gathering.

He recounted the battleground states he won in last year’s election. He said Hillary Clinton “didn’t work hard” in Michigan, eliciting boos at the mention of his opponent’s name. He resurfaced his grievances with “fake news” and repeatedly doubted that the news media would report the size of the jamboree crowd. And when he landed on the second point of the Scout Law — loyalty — Mr. Trump interrupted himself to say, “We could use some more loyalty, I will tell you that.”

More than a dozen current Republican officeholders who were Scouts declined to comment or did not respond to messages on Tuesday, but a spokeswoman for Interior Secretary Ryan Zinke, who also attended the jamboree, said that he thought the speech was “very well received” and that he “appreciated the president taking the time to talk with America’s future leaders.”

Senator Luther Strange, Republican of Alabama and an Eagle Scout, said in a statement: “President Donald Trump was absolutely right when he said that the United States has no better citizens than its Boy Scouts and it’s wonderful that he has appointed so many Eagle Scouts to cabinet level positions.’

But the president’s tone and message alarmed many other people with ties to the Scouts.

Alison Deshotels, whose 11-year-old son, Cole, has been in a Cub Scout pack in Louisiana, said her son had been wary about continuing with the organization, even though he had enjoyed camping and Pinewood Derby races. But after watching the president’s speech, he said he did not want to remain involved.

“I didn’t really agree with anything he was saying,” Cole said in an interview, “and everyone in the Boy Scouts was cheering.”

Gerald Ford is the only president to have earned the Eagle rank, but presidents of both parties have been close to the Scouts: Their signatures have been affixed to Eagle Scout certificates, they have hosted boys and leaders in the Oval Office, and many have appeared at jamborees.

“When you’re in a position to touch a young person and at least create some kind of spark that may lead them to a career in public life, don’t get distracted with your other stuff,” said Michael Dukakis, an Eagle Scout who was governor of Massachusetts and the 1988 Democratic nominee for president.

The controversy that followed Mr. Trump’s appearance presented another challenge to the Scouts, who have faced frustration and anger in recent decades for policies on gay and transgender people. Although the Scouts have rolled back many of their most controversial rules — they said this year that they would allow transgender members — the group has still struggled to cultivate cultural relevance.

The Scouts’ American affiliate said this year that it had more than 2.3 million youth participants, down from nearly 4.6 million in the late 1990s.

The Scouts have also faced separate criticism and litigation for the group’s handling of sexual misconduct, which is documented in files that the organization has held closely.

Amid the fallout from Mr. Trump’s speech, which also included an anecdote about an encounter with a New York developer at a cocktail party, some said the Scouts were not at fault.

“The Boy Scouts were not in the wrong here,” said Zach Wahls, a co-founder of Scouts for Equality, which pressured for changes to membership policies. “We should not be blaming the organization that always invites the president to speak. We should be talking about the president who took that opportunity and twisted it.”