Joe Biden recently summed up his problems getting things done.

In an America where the US Senate is split 50-50, then effectively any single senator can hold a veto over the president’s entire agenda. “Look,” laughed Biden at a CNN town hall, “you have 50 Democrats, every one is a president. Every single one. So, you got to work things out.”

It explains why the most powerful man in the world is currently struggling to get his way in Washington – and why two members of his own party, Senators Joe Manchin and Kyrsten Sinema, are standing in his way.

Such is the distribution of power that American presidents can only impose their will up to a point if Congress refuses to yield. While the White House can do much with executive orders and actions, major legislation must gain a majority of votes in the House of Representatives and Senate.

Democrats do currently control both chambers – but only just. The Senate is evenly divided between 50 Democrats and 50 Republicans, meaning that Vice-President Kamala Harris must cast the tie-breaking vote. That means all 50 Democratic senators must be on board in the face of united Republican opposition – an increasingly safe (and bleak) assumption in polarised era.

By contrast, President Franklin Roosevelt’s Democrats reached 59 seats in the then 96-member Senate, while President Lyndon Johnson’s Democrats had 68 in what by then was a 100-seat chamber. Biden is trying to match the scale of both men’s ambition with no room for error.

This is why his agenda – huge investments in infrastructure and expanding the social safety net – depends on the blessing of Manchin and Sinema in what might seem to the watching world as a case of the tail wagging the dog.

Bernie Sanders, an independent senator from Vermont defeated by Biden in last year’s Democratic primary, tweeted earlier this month: “2 senators cannot be allowed to defeat what 48 senators and 210 House members want.”

But the cold reality is that, after months of painful wrangling and concessions, Manchin and Sinema have almost single-handedly narrowed the scale and scope of Biden’s grand vision.

This week it emerged that Build Back Better plan would be halved from $3.5tn to $1.75tn, axing plans for paid family leave, lower prescription drug pricing and free community college. Sanders’s dream of including dental and vision care also failed to make the cut.

Biden insists that the framework still represents the biggest ever investment in climate change and the greatest improvement to the nation’s healthcare system in more than a decade.

He said at the White House on Friday: “No one got everything they wanted, including me, but that’s what compromise is. That’s consensus. And that’s what I ran on. I’ve long said compromise and consensus are the only way to get big things done in a democracy, important things for the country.”

But what makes it especially galling for many is that opinion polls show the eliminated measures are highly popular. Manchin and Sinema have ensured that the US is destined to remain one of seven countries – along with the Marshall Islands, Micronesia, Nauru, Palau, Papua New Guinea and Tonga – without paid leave for new mothers, according to UCLA’s World Policy Analysis Center.

Biden met late on Tuesday evening with both senators at the White House. Countless newspaper columns and hours of airtime have been expended trying to understand the motives of the two holdouts, who can effectively decide whether Democrats keep their campaign promises – with huge implications for next year’s midterm elections.

Manchin is not so mysterious. He hails from coal-rich West Virginia, a conservative state that Donald Trump won twice in a landslide, and once ran a campaign ad in which he shot a rifle at a legislative bill. He owns about $1m in shares in his son’s coal brokerage company and has raised campaign funds from oil and gas interests.

Critics accuse him of putting personal and local concerns ahead of his party, the nation and the world. The USA Today newspaper wrote in an editorial: “It’s no stretch to conclude that Democratic Sen. Joe Manchin is risking the planet’s future to protect a dwindling pool of 14,000 coal mining jobs in his home state of West Virginia.”

Sinema, however, has been described as enigmatic, sphinx-like and whimsical. In 2018 she became Arizona’s first Democratic senator for more than two decades and, despite progressive credentials and a flair for fashion statements, has taken conservative positions on several issues.

She also provokes the left with stunts such as a thumbs-down gesture on the Senate floor when she voted against raising the federal minimum wage and, on Thursday, a parody of the TV comedy Ted Lasso with the Republican senator Mitt Romney. Perhaps tellingly, Sinema raised $1.1m in campaign funds in the last quarter with significant donations from the pharmaceutical and financial industries.

In a divided Senate, where Republicans are consumed by Trump’s election lies, this duo holds the fate of Biden’s presidency in their hands. The haggling could go on for weeks more, granting Manchin and Sinema continued outsized influence and guaranteeing that their every move and word will be avidly scrutinised.

Joe Lockhart, a former White House press secretary, tweeted: “My dream? Democrats pick up a few more Senate seats in 2022 and when Joe Manchin holds a press conference to declare what he can live with and what he can’t, nobody shows up.”