I have checked for shorts on all pins and none found. I have to admit my knowledge of how a boost converter is able to switch off a load when VIN is connected to SW through the inductor may be causing some confusion for myself. Manually pulling the control pin high or low has no effect. I am going to try and write a library to control the backlight controller over 1-wire but I am not expecting that to work since I am unable to control the chip at all.
It can’t switch off the load as such. This boost topology means that the minimum Vout is Vin minus the Vf of the Schottky diode. (=Boost converters cannot reduce the voltage to less than the input voltage minus the diode drop.) The idea here is that with a typical e.g. 12-36V string of LEDs, the Vin-Vf will be much too low to light the LEDs, so boosting is necessary to get anywhere near a voltage that will light the LED string.
So maybe this is the issue: in your schematic you show a single LED. I initially assumed this was shorthand for some higher-voltage LED module that puts multiple LEDs in series. But if it is, in fact, a single LED (or multiple in parallel) with an LED Vf of e.g. 3.2V for white, then when you power it with 3.3V, it’s never actually boosting the voltage and thus cannot control the LED. Assuming a 0.3V drop in the Schottky, that means 3.0V is reaching the LED, which is easily enough to light a white LED.
FWIW, in a project I designed, the LCD backlight (4x 3.2V white LEDs in parallel) was controlled by one channel of an NXP PCA9634 LED driver. I like that model because it has a very high 97KHz PWM frequency, so no flicker whatsoever. But it’s overkill for a single channel, since it’s an 8-channel chip. The ESP32 can do high-speed PWM, so if your backlight is a single or parallel LED, you could just use a MOSFET to PWM the 3.3V to the backlight.