I'm pretty sure cell towers all use gps for time synchronization, which is necessary for the network to work.

I always trust my cell phone time.

Cell towers need synchronized clocks? For what?

I'm pretty certain that CDMA towers, at least, had exceptionally accurate clocks for something related to the technology (synchronizing the chip code maybe? it's a guess).

Not sure if this applies to GSM as well, but CDMA used to be an acceptable clock source for a Statum 0 clock source.