As far as I can tell those boost converters have 2 flaws that I'm not fond of.
One, the output is pulsed, as all boost converters will be unless an output inductor is brought in. There's a cap there to try to filter the output but these aren't very effective with LEDs with a low dynamic resistance; unless the cap is very large a small amount of voltage ripple makes the LED current vary a great deal.
Two- and this is what I think the primary problem is- I don't think they compensate for pulse width. If the battery is fully charged the output may be on for 80% duty cycle and if it's running down the duty cycle may be 40%. So even if the inductor is large and has virtually constant current, the average current the load sees is inductor current * duty cycle, so it is down by 50% when the battery runs down in this example.
I wonder if anybody makes a part that uses a ramp generator to compensate for the inductor current? So the vref it compares the shunt voltage with increases while the inductor remains in the charge phase?