The driver chip is likely pretty fast. These are intended to be used for driving power FET's in switch-mode power supplies, operating at a couple dozen kHz up to units of MHz. Whenever I hear about too much consumption in a fast circuit, self-oscillation is among the theories I need to test for. Hence my further comments:
1) you haven't shared a photo of your PCB with us.
2) do you have access to an oscilloscope?
So you're on a solderless breadboard...
Speaking of oscillations, your driver chip model is non-inverting, which alone hints that there's room for positive feedback... but the "energy coupling" needed for oscillation can happen in various cunning ways. Long power suppy leads can radiate. The output FET switches apparently can "pack quite some punch" = are fast AND strong, can yield quite a bit of dI/dt.
I recall being slightly surprised by a stepper motor driver chip, that did not stay open for a long time. It had some sort of protection where it would shut off the output after some fixed period of input staying high. Like a "minimal PWM frequency requirement" from the PWM controller. But I tend to believe that this is not your problem. The datasheet mentions "local ON/OFF switch" as a particular application. This would suggest standard continuous operation (DC, active state unlimited by time).
If your goal is to test the driver chip, I wish you good luck getting to an oscilloscope. If your goal is just to blink a LED, you don't need the FET driver. I suggest that you use a small FET with V_gs_th low enough to accept drive by 5V TTL. I'd use a BS170 or 2N7000TA (note that they have different pinouts in the TO92 package). Actually your SUP70030 might be happy with 5 Volts at the gate too, but seems a bit of an overkill for just an LED :-)