You're making a wrong assumption that the efficacy goes up with higher power level. The opposite is true, to any meaningful power levels the efficacy decreases whenever you increase the current.
PWM is used because it's very easy to implement. If you set your current to the maximum you want to use, you can have linear brightness control by just adjusting the duty cycle. Adjusting current has nonlinear response requiring calibration table if absolute accuracy is important (it often is not).
As you can see from this 1W white led Lumen vs current curve, doubling the current does not double the light output. Whether this is important depends on your application. If you're dealing with a >1kW advertising backlight, the electricity bill easily exceeds the up front cost of the display module. There are also thermal considerations, with better efficacy you have less waste heat on your system.
To make things worse is that efficacy drops even more with higher junction temperature. This graph shows ambient temperature but essentially junction temperature works in similar fashion. They're just being difficult about it. Now PWM will average the heat output but still, worse efficacy requires higher average current which means higher junction temperature..
One downside of a PWM is that the load is nasty from SMPS point of view, you're effectively imposing constant radical transients to the poor thing. At the very least you need a large output capacitor to buffer the voltage dips and peaks at edges.
A problem with constant current driving is that it's more complicated, especially if you want adjustable output current. There are further complications with local dimming applications as Vf varies with output power level so your current regulator has to dissipate the difference.
Edit added bit about junction temperature.