If only it were possible to receive such a broadcast without needing wifi.

It is:

https://en.wikipedia.org/wiki/Radio_clock

https://en.wikipedia.org/wiki/WWVB

I've had several that set automatically after plugging in, without the need for wifi.

It is possible to find real WWVB clocks, but note that most of the cheap "auto set" ones just have a lithium battery to remember whatever time was set at the factory.

It'd be cool if WiFi routers could broadcast the time unencrypted, for use by passive receivers, but that would get annoying if one near you is set wrong. Perhaps there could be a trusted source of signed timestamps, and a receiver just takes the maximum.

Better is just use GPS. You can passively listen without being tracked and you get the time and location. GPS is getting cheap enough ($10/1000 qty) to be a good option for this sort of thing.

Doesn’t work super well when your clock needs an unobstructed view of the sky. How many clocks in your house have that?

It's not like it needs a solid fix though. It just needs a single satellite signal to get through, which is way more reasonable.

I have a higher end GPS module. Building I'm in is corrugated steel sided with a metal roof. After being on for an hour the coordinates show... my desk. Far as I can tell.

GPS transmit power is 44db with a 13 db antenna gain. I think. And -140 db sensitivity (typical) So close to 200db of link margin. And the transmitter isn't in clutter either.

Don't mobile networks broadcast the time already?

Yeah, I was continuing the GP's sarcastic comment about WWVB etc.

GPS clocks do just this - https://timetoolsltd.com/gps/what-is-the-gps-clock/

So do non-GPS plain old longwave and shortwave radio clocks - https://en.wikipedia.org/wiki/Radio_clock

GPS clocks works inside a bedroom without a clear view of the sky?

My GPS receiver works just fine inside, even with the window blinders closed. I suppose it might not in a windowless room.

GPS clocks respect local timezone and DST rules?

I have several in my house. NTP amd WWVB are two different things.