Equilibrium, not a target
This is not the pressure that carbonates the beer fastest. It is the pressure at which the beer stops absorbing CO₂ — the point where gas going into solution and gas coming out balance. Set it and leave the keg alone for about a week and the beer arrives at the target on its own.
Temperature does most of the work
Cold liquid holds far more dissolved gas. The same 2.4 volumes needs about 10 psi at 3 °C and roughly double that at 13 °C. This is why a keg that tastes flat in a warm garage is not necessarily under-carbonated — it is at equilibrium with the wrong pressure for that temperature.
It also means the serving temperature and the carbonating temperature have to be the same, or the beer will drift. If you carbonate cold and then serve warm, CO₂ comes out of solution and you get foam.
Carbonating faster
You can push higher pressure to speed things up — 30 psi with the keg rocked by hand will carbonate in minutes rather than days. The catch is that you must then drop back to the equilibrium pressure, or the beer keeps absorbing and overshoots. Over-carbonated beer takes far longer to fix than under-carbonated beer takes to finish.
Line balance
If the equilibrium pressure pours nothing but foam, the problem is usually the dispense line, not the carbonation. Beer line has a resistance per unit length; you need enough of it to drop the keg pressure to near zero at the tap. Too short a line and the beer comes out under pressure and breaks out of solution in the glass.