I agree that the referenced page is the relevant one. (
Link)
The way I read it, the text quoted above refers to people crossing a land border. (i.e. already inside the Schengen area)
There is a specific section further below on that page under section "Information for international travellers" and then "transiting Switzerland".
The essential part there is:
It is possible to stop over at one of the Swiss airports, as long as you do not leave the transit zone of the airport and you can prove that you can continue your journey to your destination country.
If you are not subject to the visa requirement, you will be granted entry to the Schengen area, provided no entry restrictions apply.
So the decisive part that does or does not allow you to go landside is if a restriction exists, which will mostly be based on the your passport.
Based on what I read (still on the same linked page) under "questions on border closures and exceptions" / "what exceptions apply to this border closure", my understanding is that a US passport will not give you entry to Switzerland, because the rule that "You are simply travelling directly through Switzerland to another country that you are permitted to enter." does not require that you have to go into the city and come back.
Not really crystal clear, I'm afraid, but I would not expect to be granted entry with a US passport, even if it's just for overnight.
There's a thread in the covid forum where Sophialite tried to enter the Schengen area in ZRH on her way from the UK to Sweden, but that was denied.
Link
I expect all lounges to be still closed, so I'd go with the
Day Rooms at the airport unless you can get a confirmation from the embassy that they'll let you in.
At least some good news: Based on the single data point I have, the flight to EWR could be pleasant. 3 Passengers in Business and 25 in Economy on ZRH-EWR.
Good luck and safe travels!