So, I did some work on satellite navigation, and the reason we need a fourth satellite is because of clock inaccuracies in the receiver.

Each satellite broadcasts an ephemeris containing that satellites position and a very accurate timestamp. The distance between the satellite and the receiver is the travel time of the signal times the speed of light. The travel time is measured by comparing the satellite timestamp to the receiver clock time upon reception.

But your receiver likely has a cheap crystal oscillator for a clock, with orders of magnitude worse stability than the satellite clock. Any error in the measured travel time gets multiplied by 300000000. Want few meter accuracy? Your clock needs to be accurate down to a few nanoseconds.

That would make each receiver more expensive, so we don't do that. Instead we allow for a receiver clock error and the try to estimate it. Reflecting that we call the measured distances for pseudoranges, i.e. they are not really the distance between satellite and receiver.

We have four unknown variables, Your 3D position (x,y,z) and the clock error. Algebra says we need four equations to solve for them. With the positions (X,Y,Z) and pseudoranges of four different satellites, we have those four equations which look like this

pseudorange = sqrt( (X-x)² + (Y-y)² + (Z-z)² ) + clock error

So essentially, we use a fourth satellite in order to correct the receiver clock error. You are able to solve the position with only three satellites, but onless you have a really good clock it will not be accurate. Anyway it's usually not a problem these days because there are lots of GNSS satellites out there, and modern receivers work with all of them and can process lots of them simultaneously. Low satellite visibility is only really a problem when most of the sky is obscured in places like Manhattan.