As President Trump pushes to reopen the economy, most of the country is not conducting nearly enough testing to track the path and penetration of the coronavirus in a way that would allow Americans to safely return to work, public health officials and political leaders say.

Although capacity has improved in recent weeks, supply shortages remain crippling, and many regions are still restricting tests to people who meet specific criteria. Antibody tests, which reveal whether someone has ever been infected with the coronavirus, are just starting to be rolled out, and most have not been vetted by the Food and Drug Administration.

Concerns intensified on Wednesday as Senate Democrats released a $ 30 billion plan for building up what they called “fast, free testing in every community,” saying they would push to include it in the next pandemic relief package. Business leaders, who participated in the first conference call of Mr. Trump’s advisory council on restarting the economy, warned that it would not rebound until people felt safe to re-emerge, which would require more screening.

And Gov. Andrew M. Cuomo of New York reiterated his call for federal assistance to ramp up testing, both for the virus and for antibodies.

“The more testing, the more open the economy. But there’s not enough national capacity to do this,” Mr. Cuomo, a Democrat, said at his daily briefing in Albany. “We can’t do it yet. That is the unvarnished truth.”

As the governor spoke, a PowerPoint slide behind him said, “WE NEED FEDERAL SUPPORT.”

At his own briefing later in the day, Mr. Trump boasted of having “the most expansive testing system anywhere in the world” and said that some states could even reopen before May 1, the date his task force had tentatively set. Twenty-nine states, he added, “are in good shape.”

From the beginning of the coronavirus crisis, lapses by the federal government have compromised efforts to detect the pathogen in patients and communities. A diagnostic test developed by the Centers for Disease Control and Prevention proved to be flawed. The F.D.A. failed to speed approval for commercial labs to make tests widely available. All of that means that the U.S. has been far behind in combating the virus.

Whether in New York City, with its densely packed 8.4 million residents, or Nebraska, with fewer than two million spread across mostly rural expanses, widespread diagnostic and antibody testing will be crucial for determining a number of factors: How many in a community are infected but asymptomatic? Who has the protective antibodies that might allow them to go about their lives without fear? Are workplaces and schools safe?

“It is great that we are flattening the curve,” said Dr. Mark McClellan, director of the Margolis Center for Health Policy at Duke University, who worked in the George W. Bush administration and is advising state and federal policymakers on the virus response.

“But for this next phase, where we are really aiming to detect and stamp out smaller outbreaks before they get so big, testing is critical for that,” he said. “So we have to plan ahead now for much larger capacity.”

By the end of May, he added, “we will maybe be up to two million tests a week, but we are definitely not at that level now.”

Nationally, an average of 145,000 people have been tested for the virus each day over the past week, according to the Covid Tracking Project, which reported a total of nearly 3.1 million tests across the United States as of Tuesday night.

State health officials and medical providers around the country say they are unable to test as many people as they would like. Many of them say the biggest challenge is getting not the diagnostic tests themselves but the supplies to process them, including chemical reagents, swabs and pipettes. Manufacturers are facing a huge global demand as every country fights the pandemic, with many attempting the widest-scale testing they have ever undertaken.

“We’re at a really critical juncture and the supply chain has not yet caught up,” Scott Becker, chief executive of the Association of Public Health Laboratories, said on Wednesday.

Yet even as people waited hours for drive-through testing in California, Florida, New Jersey and elsewhere, some laboratories reported having ample capacity.

Two weeks ago, officials at University of California San Diego Health rushed to scale up testing, setting up a second laboratory devoted only to Covid-19. “You know the saying, ‘If you build it, they will come’?” said Dr. David T. Pride, director of the molecular microbiology laboratory there. “We built it and nobody has come. ” He said confusion over which laboratories were accepting tests, and “convoluted” systems connecting providers to labs, meant his facilities were running about 200 to 300 tests per day when they could handle 1,000.

Quest Diagnostics, one of the nation’s biggest testing laboratories, said on Wednesday that it could now process more tests than it was receiving, and that it was reaching out to state health departments, doctors and nursing homes. After dealing with backlogs for weeks, the company said it was returning results in less than two days for ordinary patients, and in less than one day for priority patients.

In Nebraska, as of Wednesday afternoon, 11,757 people had been screened for Covid-19, and of those, 901 were positive, according to state health data.

Peter C. Iwen, director of the Nebraska Public Health Laboratory, said that chemicals and equipment needed to run the tests were going to places like New Orleans and New York. “We’re trying to compete with those people, and we’re just not getting the reagents sent to us,” he said in an interview with the Omaha television station KETV.

The nonprofit Community Health of South Florida is operating three drive-through sites in the Miami area and the Florida Keys, where it has provided free testing to 1,300 people.

Tiffani Helberg, the group’s vice president for communications, said a tight supply of testing swabs as well as staffing numbers meant the nonprofit was not screening as many people as it would like.

“Is it a struggle every day? Absolutely,” she said.

The lack of testing is hitting minority communities especially hard, according to Dr. James E.K. Hildreth, president and chief executive of Meharry Medical College in Nashville, one of the nation’s largest historically black medical schools.

“Testing should be a priority for vulnerable populations — that would be prisons, nursing homes, assisted living facilities and, last but not least, minorities and disadvantaged communities,” said Dr. Hildreth, an infectious disease expert. “Because in those communities, we know there are many individuals with underlying conditions, and they are more likely to get severe disease and die.”

But even as short supplies are limiting who can get tests, some laboratories say they have extra capacity.

The American Clinical Laboratory Association, a trade group representing large diagnostic companies like LabCorp and Quest, has recently reported a dip in the daily testing volumes of its members. On Monday, its members processed 43,000 tests, the lowest number since March 20. At one point in early April, members were processing more than 100,000 a day.

“They are reaching out to providers to make sure they know that we have more testing capacity,” said Julie Khani, president of the lab association.

But even as testing for active coronavirus infections is struggling to meet demand, public health officials and major laboratories say they are gearing up for the next wave: antibody testing. A well-designed antibody test will detect whether someone has been exposed to the virus and generated an immune response, and whether the person may be protected from further illness.

“Antibody testing is not a cure-all,” Gov. Doug Ducey of Arizona, a Republican, said on Tuesday as he announced a partnership with the University of Arizona to provide antibody tests for 250,000 health care workers and emergency responders. “But learning more about it is an important step to identifying community exposure, helping us make decisions about how we protect our citizens and getting us to the other side of this pandemic more quickly.”

Most of the available antibody tests can say only whether someone has antibodies, not how many they have or how powerful they are at fighting the virus. Many of the tests are also flawed and signal the presence of antibodies even when there are none. The F.D.A. has granted emergency approval to three companies to begin selling the tests, but dozens more have entered the market after the agency loosened the guidelines in March.

“We have to to make sure it’s an accurate test with good specificity,” said Dr. Rachel Levine, Pennsylvania’s health secretary. “And we really need to know that antibodies are truly protective and how long-lasting they are.”

Dr. Jon R. Cohen, the executive chairman of BioReference Laboratories, which is processing tests at drive-through sites in New York and New Jersey and other locations around the country, said he was still evaluating different antibody tests but planned to begin offering them soon. Other large laboratories said the same.

“It’s a huge factor, we believe, in terms of people regaining confidence and jump-starting the economy,” he said. “To me, it’s an absolute moral imperative.”

Reporting was contributed by Emily Cochrane, Sharon LaFraniere, Apoorva Mandavilli, Jesse McKinley, Rick Rojas and Brian M. Rosenthal.