Hi,
You need some negative offset in your brightness regulation.
Opamps are no ideal amplifiers.
One disadvantage is that the inputs have a low +/- some mv input offset voltage. If the gate driving opamp has only some mv of positive offset, then even at zero input voltage at + input it wants to regulate. So it drives the fet slightely on so that a small current flows through the leds and cause the some mV on the - input.
This is what happened in your first description.
The next problem with opams is that they can not regulate its output to true 0V, if the lower supply voltage is at 0V.
Rail to rail output opamps are optimized on this.
Therefore the bnc buffering opamp can go down to only some mV on its output. The remainig mV cause the leds to light woth low brightness.
Additionally the human eyes are very sensitive on light. Even if there are only some mA with 1A rated Leds you will recognize it.
To avoid your problems you can add a negative offset to your current regulation. This can easily be done with two resistors.
Place one 100 Ohms resistor in the line from 1Ohms shunt to -input of opamp.
And place a 20k resistor from - input to +5V. Now you have about -25mV offset.
Hope this helps.
Btw. Paralleling of leds without series resistors may cause difference in brightness.
Linear regulation will cause power loss and heating. Switching regulation can imrove on this.
Klaus