Cases of Covid-19 will continue to surge worldwide due to the Omicron variant, the US chief medical adviser, Anthony Fauci, said on Sunday, warning Americans not to get complacent amid reports that the variant is less harmful than others.

“If you have many, many, many more people with a less level of severity,” Fauci told ABC’s This Week, “that might kind of neutralise the positive effect of having less severity.

“We’re particularly worried about those who are in that unvaccinated class ... those are the most vulnerable ones when you have a virus that is extraordinarily effective in getting to people.”

Fauci also welcomed Donald Trump’s endorsement of Covid-19 vaccines and boosters, saying: “We’ll take anything we can get about getting people vaccinated.”

But Trump prompted rebarbative anger among supporters and amid a huge case surge, with knock-on effects feared for the economy and schools, Fauci also admitted the US had “to do better” on providing access to testing.

Speaking to Axios, Fauci said it was “conceivable that sooner or later everybody will have been infected and/or vaccinated or boosted”.

“When you get to that point,” he said, “unless you have a very bizarre variant come in that evades all protection – which would be unusual – then I think you could get to that point where you have this at a steady level.”

But he also suggested fourth shots might yet be needed. On ABC, he was asked why “we still don’t have affordable tests widely available to anybody who needs it”.

“If you look at the beginning of the [Biden] administration,” Fauci said, “… there were essentially no rapid point-of-care home tests available. Now, there are over nine of them and more coming. Production has been rapidly upscaled.

“… But the situation where you have such a high demand, a conflation of events, Omicron stirring people to get appropriately concerned and wanting to get tested as well as [a] run on tests during the holiday season – we’ve obviously got to do better.

“I think things will improve greatly as we get into January, but that doesn’t help us today and tomorrow. So you’re right, [access to testing] is of concern.”

Another leading public health expert said he did not think the case for possible fourth vaccine shots needed to be made right now.

“If we need it I think our health system is prepared,” Ashish Jha, dean of the Brown University School of Public Health, told Fox News Sunday. “But let’s actually talk about whether we need it or not. And at this moment, based on the data I’ve seen, I’m pretty skeptical that we’re gonna need a fourth shot.

“Part of the question is that we have to ask ourselves what are we trying to do? Are we trying to block every single infection? Maybe that’s our goal. If that’s our goal then yes, maybe we need a fourth shot. Or are we just trying to prevent serious illness and death? Which, of course, I think should be our primary goal.

“So I’m pretty unconvinced at this moment that we need a fourth shot … let’s get a lot more data before we even really start seriously thinking about it.”

Jha also said school closures – feared by many parents – should not be increasing.

“We know how to keep schools open,” he said, “we know how to keep them safe. This really shouldn’t even be on the table. I’m disappointed to see this happening.

“We know that for kids being in school is the right thing for them, for their mental health, for their education. And we have all sorts of tools to keep schools open so I don’t really understand why school districts are [closing schools].

“… There could be times when you have such severe short staffing shortages that it may be hard to keep schools going. That really should be the only context I think at this point.”

More than 816,000 have died from Covid in the US but resistance to vaccinations and other public health measures remains strongest in states and counties that voted for Trump. On ABC, Fauci was asked if he thought the former president’s supporters would listen to his support for vaccines.

“Well, I certainly hope so,” he said. “We’ll take anything we can get about getting people vaccinated.”

But Fauci also said he was “dismayed” when Trump followers in Dallas booed him for supporting vaccines.

“I was stunned by that,” he said. “I mean, given the fact of how popular he is with that group, that they would boo him … tells me how recalcitrant they are about being told what they should do.

“I think that his continuing to say that people should get vaccinated and articulating that to them, in my mind is a good thing. I hope he keeps it up.”

Trump also backed vaccines in an interview with the conservative commentator Candace Owens, saying: “The vaccines work … the ones who get very sick and go to the hospital are the ones that don’t take the vaccine … and if you take the vaccine, you’re protected.”

On Instagram, Owens said Trump was backing vaccines because he was “old” and “came from a time before TV, before internet, before being able to conduct … independent research”.

Last week, after Biden recognised his predecessor’s efforts to develop vaccines, Trump said he was “appreciative”. Biden also commended Trump for receiving a booster, saying it “may be one of the few things he and I agree on”.

On Sunday, Vice-President Kamala Harris was asked on CBS’s Face the Nation if the unvaccinated were to blame for the Omicron surge.

“I don’t think this is a moment to talk about fault,” Harris said.

But she added: “It is clear that everyone has the ability to make a choice to save their lives and to prevent hospitalisation if they get vaccinated and if they get the booster. And so I urge people to do that.”