GPS satelites are just fancy atomic wirstwatches hanging on Earth orbit

No, more is preferred, but the way the signals are designed, some positioning slowly works with only two satellites.

s/signals/receivers

I guess receivers that also measure angle to satelites do have 4 constants with only 2 satellites to get 4d solution(or rather 2 solutions, one of which is in future, while other is in the past, not sure which one is correct). Or maybe try to do some wierd math shit with Doppler shift. I was talking about original(cheap and easy) way receivers were solving coords with 4 latencies.

If we are really pushing the limits here, then receiver that knows it doesn't move relative to Earth can get coords from one satelites, but this is just speculation. And it may require atomic clock.

Remember when GPS was slow and always a few meters off? Part of that was they were bad at or could not acquire more than two signals.

This doesn't sound like reason for it. Slow start? Receiver first needs to receive ephemerides and almanac to be able to solve position. Quoting wikipedia article on A-GPS:

Every GPS device requires orbital data about the satellites to calculate its position. The data rate of the satellite signal is only 50 bit/s, so downloading orbital information like ephemerides and the almanac directly from satellites typically takes a long time,

Almanac can be stored on device for a long time to be used later in next start. It's called warm start. Ephemerides don't last as long. Start when they aren't stale called hot start.

Now goes quote from another article

An ephemeris is valid for only four hours, while an almanac is valid–with little dilution of precision–for up to two weeks.

So bright minds thought "what if almanac and ephemeris" will be downloaded from the internet? And this is how A-GPS was born.

Now about precision. Few meters is normal precision for normal GPS. Getting better precision requires very scary math hiding behind Differential GPS and additional correctional data. Not that normal GPS doesn't have scary math.

Lastly, about two satelites: old phones had one antenna per band(and usually only one band), so they did require at least 4 satelites.

I did not misspeak. I linked to a-gps which explains how the system works.

You do not need to transmit or receive cell data to receive a GPS signal. If you use GPS isolated, your location isn't leaked.

But that's irrelevant because if you use cellular data, the cell tower knows approximately where you are.

I have been using GPS since before cell phones had data. My boss back in 1994 brought in the man who wrote the accuracy encryption part of gps (that has since been turned off) to give a talk on the encryption that used to be in GPS to myself and my coworker just for the fun of it because they were old friends.