Why we consider second derivative w.rt price but only first derivative w.r.t time and volatility

What is the reason (better if it is intuitive, and not too math heavy), that when we talk of Greeks, we consider second derivative with respect to price (gamma), but only first derivative with respect to time (theta) and volatility (vega).

At least, most brokerage platforms only publish values for these Greeks. Why not consider the second derivative with respect to time and volatility? Are they not important?

Consider, instead, $\frac{\partial \theta}{\partial t}$, i.e. the second derivative of price w.r.t. time. What sort of financial application might this have? By itself, $\theta$ tells us the rate of change of the option price w.r.t. time, and so $\frac{\partial \theta}{\partial t}$ would be the "acceleration" of the price. While this has a nice physical meaning, it isn't of much use from a hedging perspective, in that it isn't explicitly used to calculate hedges.