WASHINGTON — Joseph R. Biden Jr. has proposed harnessing the broad powers of the federal government to step up coronavirus testing, with a public-private board overseeing test manufacturing and distribution, federal safety regulators enforcing testing at work and at least 100,000 contact tracers tracking down people exposed to the virus.

The presumptive Democratic nominee’s plan, laid out in a little-noticed Medium post, stands in stark contrast to President Trump’s leave-it-to-the-states strategy, detailed in an 81-page document released over the weekend. And it presents voters in November with a classic philosophical choice over the role they want Washington to play during the worst public health crisis in a century.

With more than 100,000 Americans already dead from the coronavirus and at least 1.7 million infected, testing has emerged as a major campaign issue. Polls show that most people want better access to testing and believe that it is the job of the federal government. Like Mr. Biden, Democrats running for Congress have seized on testing as a prime example of what they view as Mr. Trump’s incompetent response to the crisis.

In Michigan, Senator Gary Peters, an incumbent Democrat, tells viewers in a TV ad that “our workplaces need to be safe” and “that means more testing.” In Colorado, an ad for Senator Cory Gardner, an incumbent Republican, begins with footage of a news anchor saying, “Coronavirus tests are coming to Colorado from South Korea because of Senator Cory Gardner.”

In Maine, Sara Gideon, a Democrat running to unseat Senator Susan Collins, is airing an ad in which she says that “the federal government needs to expand testing, which is critical to keeping us safe.” In Washington, Speaker Nancy Pelosi held a news conference on Tuesday to attack the Trump plan as insufficient.

“Mr. President, take responsibility,” Ms. Pelosi declared, adding, “That’s what the president of the United States is supposed to do.”

Beyond the slogans and congressional calls for a national testing strategy, Mr. Biden’s plan, laid out late last month as he struggled to grab voters’ attention, begins to flesh out what such a strategy would entail.

Harking back to the War Production Board created during World War II by Franklin D. Roosevelt, the former vice president proposed a “Pandemic Testing Board” to oversee “a nationwide campaign” to increase production of diagnostic and antibody tests, coordinate distribution, identify testing sites and people to staff them, and build laboratory capacity.

Testing, he and his advisers wrote, “is the springboard we need to help get our economy safely up and running again.”

Mr. Biden said he would do what the Obama administration did during the H1N1 pandemic of 2009 — instruct the Occupational Safety and Health Administration, which regulates workplace safety, and the Centers for Disease Control and Prevention to issue detailed guidance for how employers should protect their workers, including testing, campaign advisers say. OSHA would enforce compliance.

Under Mr. Trump, OSHA has issued Covid-19 guidance for employers that is “advisory in nature and informational in content” and does not mention testing. The C.D.C.’s interim guidance for employers says only that companies “should not require a Covid-19 test result” or a doctor’s note to grant sick leave or to determine whether employees can return to work.

Mr. Biden would also create a federal entity: the U.S. Public Health Jobs Corps, a force of at least 100,000 people, including AmeriCorps and Peace Corps volunteers and laid off workers, to trace the contacts of those who test positive for the virus. It would also “become the permanent foundation” of a service that would address other public health priorities like the opioid epidemic.

Republicans argue in favor of a more localized response led by state governments. “With support from the federal government to ensure states are meeting goals, the state plans for testing will advance the safe opening of America,” says the Trump administration’s 81-page Covid-19 Strategic Testing Plan, prepared by the Department of Health and Human Services.

Senator Lamar Alexander of Tennessee, the chairman of the Senate Committee on Health, Education, Labor and Pensions, called Mr. Biden’s idea a “typical Democratic response.”

“There’s a big difference between what’s going on in Queens, N.Y., and rural Tennessee, and the governors know best what to do,” he said, adding, “Every time you have a national problem, whether it’s education or health, the instinct of Democrats is to say, ‘Let’s solve it from Washington,’ and my instinct and that of Republicans is that this is a country that works state by state, community by community.”

Some public health experts, including those who advise the Biden campaign and some who do not, say that is a false dichotomy. The federal government could and should cooperate with and support the states, and also take a more aggressive role, they say, particularly in a chaotic environment where a global shortage has left governors — and now employers — competing for scant supplies of test kits and wondering how best to use them.

“Every university, every employer, every organization is struggling to figure out how to use testing to create a safe environment,” said David A. Kessler, a Biden campaign adviser who was the commissioner of food and drugs under Presidents George Bush and Bill Clinton.

“If you’re Amazon,” he added, “you can hire people to put in place testing systems to help assure the safety of your work force, but not everyone can do that. Why are we reinventing this firm by firm, school by school, employer by employer?”

Congress required Mr. Trump to provide a national testing strategy in the $484 billion stimulus package it passed last month and required the states to submit plans to the federal government for approval. But Democrats on Capitol Hill say the strategy the Trump administration offered over the weekend falls far short of what they envisioned.

Experts say there should be two main components to a comprehensive national testing strategy: a centralized effort to acquire test kits and distribute them, and clear guidance on how to use them.

Andrew Slavitt, who was the acting administrator of the Centers for Medicare and Medicaid Services under President Barack Obama and has provided advice to the Trump White House during the pandemic, said one reason for the government to control the acquisition of coronavirus tests was that commercial labs were increasing their prices to as much as $140 a test.

“In this laissez-faire policy, there are scarce resources, and whoever has the scarce resources gets to charge what they want, and the states all get to bid and now the employers are bidding,” he said. “The consequences of this are to make the distribution much more costly, much more uneven.”

As for how to use the tests, Republicans say such plans are best developed state by state, community by community. But with a virus that respects no borders, Democrats insist that a national standard is essential.

“We would say, if you want to reopen a school, then you have to test so many kids per day; they have to be retested every so often,” said Representative Frank Pallone Jr. of New Jersey, the chairman of the Energy and Commerce Committee, adding: “The same thing with employers. How many people have to be tested before it’s safe to go back to work? How often do they have to be retested?”

Polls show that voters tend to favor a prominent role for the federal government. In a Pew Research survey released this month, 61 percent of Americans said coronavirus testing was mostly or entirely the responsibility of the federal government, not the states.

A Fox News poll released last week found that 63 percent of registered voters viewed the “lack of available testing” as a “major problem.” Just 12 percent said it was not a problem at all. Voters said they trusted Mr. Biden to do a better job on health care than Mr. Trump by a 17-point margin and favored Mr. Biden on the handling of the pandemic by nine points over Mr. Trump.

In a CNN poll earlier in May, 57 percent of Americans said the federal government was not doing enough to address the limited availability of coronavirus testing.

“When Americans hear Trump talking about testing not being his responsibility, the takeaway is that he’s just passing the buck,” said Geoff Garin, a Democratic pollster.

While Mr. Trump has repeatedly said that anyone who wants a test can get one, that is not true in many parts of the country. It is true in Tennessee, where Gov. Bill Lee, a Republican, has decided that the state will pay for testing. “When in doubt, get a test,” he said on his Facebook page, adding, “Aggressive testing is key to our reopening strategy.”

And while Mr. Trump has emphasized the number of people who have been tested — more than 15 million Americans, as of Monday — experts say the more important metrics are what percentage of the population has been tested, what percentage of tests come back positive and how those tests are deployed.

“Instead of focusing on what we need to do as a country to keep ourselves and our populations and especially our vulnerable people safe, and saying let’s come up with the right testing strategy and make sure we have enough tests to implement it, we’ve just been fighting about the number of tests,” said Ashish Jha, the director of the Harvard Global Health Institute.

Testing has been a confounding issue for Mr. Trump since the early days of the pandemic, when sloppy laboratory practices at the C.D.C. caused contamination that rendered the nation’s first coronavirus tests ineffective, delaying the rollout. The country never quite caught up.

Countries that had aggressive early testing campaigns — South Korea, Singapore, Taiwan and Germany, among others — have largely controlled their epidemics. Before the pandemic there were 12 direct flights between Taiwan and Wuhan, China, its epicenter, each week.

Mr. Trump’s travel ban “led us to believe that we had shut the barn door when there was a flood of virus coming into our country from multiple direction,” said J. Stephen Morrison, who runs a global health program at the Center for Strategic and International Studies. He added, “We didn’t have a testing system and we didn’t want one.”

Emily Cochrane and Thomas Kaplan contributed reporting from Washington, and Giovanni Russonello from New York.