None of the 93 homeless people who tested positive for the coronavirus in San Francisco’s largest shelter showed serious symptoms of the disease, lending urgency to the need for widespread public testing and emphasizing the importance of “contact investigations” the city is now ramping up.

The COVID-19 cluster revealed Friday at the Multi-Service Center South shelter by public health officials is the biggest in the United States involving homeless people aside from a shelter in Boston, where about 200 people tested positive earlier this month. All of them were also asymptomatic, officials there reported.

Public health officials said they wish they could test everyone, and not just people in shelters, but there aren’t enough tests and equipment to do that. So they are doing what every other county in the Bay Area is generally doing — testing those who show symptoms or are particularly vulnerable because of exposure to the disease.

On Wednesday, the city announced a process to help health workers quickly test and identify people newly infected with the coronavirus and then track down anyone with whom they may have had contact. It’s that method, called contact investigation or contact tracing, that led to the discovery of the cluster at the 340-bed MSC South, where the population had been reduced to around 150 people to create physical distancing.

Rachael Kagan, spokeswoman for the San Francisco Department of Public Health, said after two people tested positive at MSC South around April 5, health teams launched an investigation and began testing. As the numbers of people who had contact with the two positive cases grew, and then known contacts with those people grew, the decision was made to test everyone at the shelter.

“No one was seriously ill at the time they were tested,” Kagan said.

Ten staff members also tested positive. As of Thursday, three of the total from the shelter had developed symptoms serious enough to be hospitalized, but none was in intensive care, city officials said.

“It’s pretty straightforward,” Kagan said of the tracing and testing at MSC South and other shelters — including the Division Circle Navigation Center, which on April 2 reported the first positive COVID-19 case in the city’s homeless population.

“It’s not to say we wouldn’t test a whole shelter if it’s warranted,” she added. “But at this point with the supplies and capabilities we have, we are prioritizing vulnerable groups, prioritizing outbreaks and following the evidence from contact investigations to guide our decisions.”

She said the Division Circle testing had revealed no other cases in that shelter.

“We are continuing to expand testing throughout the city,” Kagan said, “and as we get more equipment, we look forward to expanding even more.”

The inability to test everyone in shelters has frustrated homeless people and their advocates for weeks.

“The best thing would be if we could test everyone,” said Jennifer Friedenbach, head of the San Francisco Coalition on Homelessness. “We don’t know if what is happening at MSC is at other shelters too, and we won’t know until we test. And if we’re not going to test, then really what’s available to us is moving people into hotel rooms as a preventative measure.”

The city intends to lease as many as 8,250 hotel rooms, and most of those are for people experiencing homelessness, with the rest going to frontline crisis workers and people living in crowded settings like SRO hotels.

San Francisco Chronicle staff writer John King contributed to this report.

Kevin Fagan is a San Francisco Chronicle staff writer. Email: kfagan@sfchronicle.com Twitter: @KevinChron