I'm curious why we don't just use GPS's UTC time broadcast to set time? I understand that using GPS to track satellites and find location is power hungry, but I can't imagine that simply receiving a time and setting a clock is any more power hungry than this system.
GPS determines a solution for position and time simultaneously. It requires 4 satellites to solve for 4 unknowns (X, Y, Z, t). If you know your position you can solve for time using only 1 satellite, but a watch is always moving so it's position won't be known.
There's no need to 'solve' for time - it's included in the raw data packet from any single satellite.
As any 'solution' carries an inherent error margin, the desire for an excess of sats doesn't stem from solving for time but the reduction of error.