LED Backlight driver output current

This was going to be a comment, but there's too many of them to fit:

Just to start:

1. The ground pad of your chip has no vias to provide cooling or low inductance returns.
2. Your MOSFET is not a great replacement, but might work okay enough below 10A peak currents.
3. Your inductor is drawn as a capacitor on the board, that's bad design form to start and downright confusing if you want actual help. Same with the block you drew as your MOSFET in the Schematic to then add insult to injury by using a Bipolar board package for it (B-C-E!!!). 3.b. Same goes for the schematic component for the diode.
4. You're using a 25miliOhm current sense resistor with no Kelvin Connection (seemingly designed on in a tiny 0805 package and replaced later).
5. With 25mOhm and 400mV over-current protection comparator level, that's 16A. Why? Your inductor and diode will not like that.
6. I see a solder wire go under the board in a picture and no preventative measures to protect from things like that shorting other things out.
7. The type of inductor you're using isn't particularly great at High Frequencies, which I'd expect the Chip to use for the boost circuitry. Note how the original specifically states how it's great at 1MHz and even 2MHz energy storage: That won't work that well with a huge I-Core choke.
8. There's no pictures of the bottom side of the finished PCB.
9. It doesn't even really look like the Chip's Thermal PAD is actually soldered to the board, but I'll give you the benefit of the doubt there.

Otherwise, your PDF contains only a compound print of the PCB, so who knows if there's mistakes I can't see on layers that are not the Top layer.

Oh, and (EDIT:)

10. Your BOM is not very usable. "Typical" this and "Sim Cap" that with only sometimes a part number in the footprint, and sometimes nothing usable at all makes any form of review nigh impossible.

From your first experiment using a benchtop PSU in CC mode, the output voltage reaches nearly 12V if you target is 1A. It means that Vf for your three LEDs is about 12V (4 V per LED), which sounds reasonable. On other words, to get 1 A through the LEDs you need to have a source in excess of 12 V.

However, the AL8860 is a switching regulator, it can't output 12V if Vin = 12V, it must have some room, because it has some limit on duty cycle (75% maximum recommended), and will stop working if Vout approaches Vin. Try to apply 14-16V as your input, everything should be fine then.

From AL8860 datasheet, page 7, to get 1A LED current at 10-11V level, you need at least 14 V of input voltage: