Your benchtop supply runs at 2 amp maximum and you are running a switched mode power supply with 1 amp of output current from your inductor. Your inductor has to supply an average one amp output, so its instantaneous current has to be greater than 1 amp. You are likely bumping up against your current limit.
During start-up, the inductor is being charged with a current (to ground) and this current is not going through your sense resistor. When the inductor is released, the voltage on the inductor must rise until the diodes conduct.
With a 10 uH inductor and your power supply max current of 2 amps, the maximum energy stored in the inductor is 1/2 Li^2 or 40 microjoules. Since you are charging the inductor 40,000 times per second, you are producing only 1.6 W (1 joule per second = 1 watt). This is probably enough to give you the low brightness in your LEDs, but not enough to get them running at a high enough current to get your regulator started.
Think of each cycle as a small bucket of energy. If you want to run with a 2 amp limit, your bucket size is fixed by the inductor and you will need a bigger bucket (larger inductor) or more buckets per second (higher frequency.)