Why is an accurate time so important for servers and computers?
It depends. If all atomic clocks would be turned off simultaneously, right now, that would be trouble. Because everyone relies on atomic clocks. And everything is organized in such a way that it doesn't matter much if a handful or two dozen stop working, but if they all stop working, then all the time servers in the world have no source for their time anymore. 

If you announced that exactly three months from now all atomic clocks will be turned off, that would be a lot of work setting up some slightly less precise clocks, and to switch over to those less precise clocks, but cell phone service and internet would happily continue working. Even a quartz wall clock (like the one in your office) can be precise within 0.2 parts per billion. Taking 1,000 such wall clocks and carefully averaging their time would be quite accurate, and surely we can build a quartz clock for 100,000 that is a lot more precise than one that costs 100.

The problem is not the precision. The problem is that if you turn off all atomic clocks right now then we don't have anything to replace them right now. There are about 400 atomic clocks in the world. If we had 400 state of the art quartz clocks ready to be used as replacement automatically if the atomic clocks are stopped, then there would be no problem except for very few very special use cases.