Warning: I said it's going to be a dumb question 
Why is so much emphasis put on them? They represent instantaneous snapshots of continuous functions of time-to-expiration, implied volatility, and underlying price. Do people really find them more useful to evaluate positions than a graph of the above variables?
For example, here is a risk chart from Livevol X courtesy of Lightspeed. https://www.lightspeed.com/wp-content/uploads/2015/05/livevol-risk-chart.jpg. (You could imagine how one might transform this into a 3D graph.) I compare that to most other software, which will show you your position delta, gamma, vega, and theta. But I don't understand how those snapshots are all that useful compared to seeing how they will change based on the underlying inputs. Personally I find the graph much more useful. Do you find them useful for quickly evaluating and choosing among a number of different positions?

Why is so much emphasis put on them? They represent instantaneous snapshots of continuous functions of time-to-expiration, implied volatility, and underlying price. Do people really find them more useful to evaluate positions than a graph of the above variables?
For example, here is a risk chart from Livevol X courtesy of Lightspeed. https://www.lightspeed.com/wp-content/uploads/2015/05/livevol-risk-chart.jpg. (You could imagine how one might transform this into a 3D graph.) I compare that to most other software, which will show you your position delta, gamma, vega, and theta. But I don't understand how those snapshots are all that useful compared to seeing how they will change based on the underlying inputs. Personally I find the graph much more useful. Do you find them useful for quickly evaluating and choosing among a number of different positions?