The transcription in the question is an accurate representation of her speech, which is available on the C-SPAN vido attached to this tweet
The pause in her speech between "herd immunity" and "quackery" could be interpreted as apposition or it could be enumeration - that is, referring to something else, e.g. the promotion of hydroxychloroquine as a treatment (also done in the Oval Office), not necessarily that she's saying herd immunity is quackery. (The FDA eventually withdrew authorization for HCQ as a treatment for COVID-19.)
For more context, right before that sentence she says
We couldn't pass legislation until now because the Administration simply did not believe in testing, tracing, treatment, wearing masks, sanitation, separation and the rest – scientific approach. It becomes clear to us, now, that they believed in herd immunity, quackery, springing right from the Oval Office, and not denied sufficiently by some of the CDC and the rest. It becomes clear to us, now, that they believed in herd immunity, quackery, springing right from the Oval Office, and not denied sufficiently by some of the CDC and the rest.
So, now, we have a vaccine and that gives us hope. A vaccine that is – springs from science.
(And if you also want to pick apart her previous sentence: "Pelosi falsely claims Trump doesn't believe in treatment (or sanitation)" could also be headline on a certain part of the press.)
Now for her slamming herd immunity... There are (ethical) reasons to disapprove of achieving herd immunity without vaccines for a disease like COVID-19. I'm not entirely sure how explicit Trump was about claiming he wanted this (i.e. herd immunity without a vaccine), but Pelosi is clearly attributing the latter approach to him. Nature only noted this briefly:
US President Donald Trump spoke positively about it in September, using the malapropism “herd mentality”.
Some Trump appointees and advisers were indeed more explicit in promoting herd immunity by natural infection: Paul Alexander or even Scott Atlas. The latter in particular made some claims that have been attacked as pseudoscientific, by claiming that only a relative small fraction of people need to become immune in order for herd immunity to happen:
Scott Atlas, a radiologist who has clashed with other members of the coronavirus task force over his controversial views, reportedly turned down a proposed increase in coronavirus testing by a New York University economist in September by referencing a theory that only 25% or 20% of people need the infection for the rest of the population to be protected, according to the New York Times.
What they're saying: "First of all, that 20% number is the most amazing combination of pixie dust and pseudoscience I've ever seen," Osterholm said in response to Atlas' alleged comments. "It is 50%–70% at minimum."
I guess Pelosi speaking of the "Oval Office" covers them too albeit not 100% accurately. The White House press secretary has denied that Trump really meant try to achieve herd immunity without vaccines:
White House press secretary Kayleigh McEnany told reporters last month that "herd immunity has never been a strategy" for the Trump administration's response to the coronavirus, after the president claimed that the coronavirus would disappear when people develop "a herd mentality."
... and so has
Health and Human Services Secretary Alex Azar told NBC's Chuck Todd on Sunday that herd immunity is not the Trump administration's policy to deal with the pandemic, adding, "It's a desire through vaccination to get to herd immunity, but it may be an outcome of all of those steps, but the desire is to reduce cases."
Given these fumbles (to put it charitably) from the administration, I'm pretty sure Pelosi was attacking the low-hanging fruit, i.e. that the Trump administration wanted herd immunity without vaccines.
But the Democratic-leaning press didn't really believe these rebuttals from the administration, e.g. ABC wrote in mid-October, recounting the Atlas-Fauci duels in the press:
While he has largely avoided using the phrase "herd immunity" to describe the policies he has furthered, the president's views reflect those of a small subset of scientists with a powerful ally in the White House, Dr. Scott Atlas, a neuroradiologist with no background in infectious diseases who has supplanted Fauci and other top federal public health officials as one of Trump's top medical advisers.
The scientists [that Trump trusted] argue that the United States should seek to achieve "herd immunity" to COVID-19 by allowing the natural spread of the virus through the population, while keeping only "vulnerable" groups -- like the elderly -- protected. Eventually, they say, enough people will get sick and recover -- and potentially become immune to reinfection, at least for some period of time -- that the rate of spread will diminish. [...]
"This idea that we [only] have the power to protect the vulnerable is total nonsense, because history has shown that that's not the case," Fauci said Thursday in an interview with ABC News Chief Anchor George Stephanopoulos on "Good Morning America." "And if you talk to anybody who has any experience in epidemiology and infectious diseases, they will tell you that that is risky and you'll wind up with many more infections of vulnerable people, which will lead to hospitalizations and deaths.
"So I think that we just got to look that square in the eye and say it's nonsense," Fauci said.
The Trump administration has not treated it as nonsense, though.
U.S. Health and Human Services Secretary Alex Azar met last week with Atlas and several researchers who have endorsed the ideas behind a "herd immunity" approach without labeling it as such. Atlas himself is a paid adviser to the president.
In addition to pushing for an end to coronavirus-related restrictions and insisting the country fully "open up," Trump also has repeatedly emphasized -- often with misleading statistics -- that the virus only impacts the elderly or those with pre-existing conditions. His focus on those groups jibes with "herd immunity" adherents who minimize the risk to others. [...]
The White House on Monday convened a conference call for reporters in part to draw attention to the Great Barrington Declaration, an online petition that argues in favor of achieving natural herd immunity while also using "focused protection" to safeguard the most vulnerable. The authors of the open letter boasted thousands of signatories, but Sky News found that among them were "homeopaths, therapists and fake names" like "Dr. Johnny Bananas" and "Dr. Person Fakename."
The Infectious Disease Society of America on Wednesday called the petition's ideas "inappropriate, irresponsible and ill-informed." [...]
In a statement provided by the White House, Atlas said "we emphatically deny that the White House, the president, the administration, or anyone advising the president has pursued or advocated for any strategy of achieving herd immunity by letting the coronavirus infection spread through the community." He said the Great Barrington Declaration authors "emphasized focused protection of the vulnerable and safely ending the shutdown of schools and society."
"Those specific policies are aligned with the president, who has repeatedly stated and pursued a strategy focused on saving lives by the following: aggressively protecting the vulnerable, preventing hospital overcrowding, and safely opening schools and society," Atlas said.
But John Brownstein, an epidemiologist at Boston Children's Hospital and professor of medicine at Harvard Medical School, said the Great Barrington Declaration did, in fact, back the idea of letting the virus rip through the community -- as long as the vulnerable were protected. "It's the same thing," Brownstein, an ABC News contributor, said. "I don't see the distinction." [...] "We know that every single person is a bridge to a high-risk person," he said. "It's just not practical to think you could put people who are high risk separate from the population."
So, even when not endorsing "herd immunity" explicitly, the administration, (especially since the side-lining of Fauci in favor Atlas) has been accused over and over of pursuing that "herd immunity" (by exposure) while not openly admitting it. That should put Pelosi's "herd immunity" jab in (more) context.
As far as interpreting the Pelosi quote, note that even mainstream-ish sources that lean Republican have not suggested she's speaking of "herd immunity" as a general epidemiological concept there, but rather referring to Trump's approach e.g. the National Review:
I’m relatively certain that not one person, much less “people,” in Congress has said: Hey, “I’m faith-oriented so I don’t believe in science.” [...]
But the topper is hearing Pelosi jump from asserting that Republicans have done nothing but intentionally spread the virus to reach “herd immunity” to saying, “now we have a vaccine,” as if the White House hadn’t implemented Operation Warp Speed with the intention of speeding up discovery, production, and distribution.
Granted, government gets too much credit for the COVID vaccine accomplishment. [...]
So I don't know who might have been confused and given it a different (broader) interpretation to what she said, as you suggested in your #2. (Wash Times, which you've used as your source for this q, had no commentary on that angle, only on the faith-based issue, which NR also seized upon.)