I just put the scope on pin 1: indeed, plenty of oscillations: 400ns period, 5vpp on pin 1.
When pin 3 is low (0V), then no oscillations, and the LED turn (correctly) bright.
I set up the LM393 in lab conditions, with LED and series 4k7, and then it behaves as it should: clean on or off. And pin 1 all the way to near Vcc.
Pin 3 (input +), connected to pin 4 (LM723) cannot be interfered with (because pin 4 is crucial to current sensing circuit in the LM723);
There is of course a difference in LED brightness when LM393 pin 1 is low, versus oscillating.
There is no power supply decoupling around IC1. Depending on the IC's internal circuit, a high source impedance for Vcc (and "high' in this case can be just a few ohms) can create either a positive or negative feedback path backwards through the circuit, disrupting normal operation. One chip can put enough noise on a power rail to disrupt another chip.
Power supply circuits need heavy decoupling. Each IC should have a 0.1 uF ceramic or stacked metal film cap between its power pins, as close as possible with short leads. In parallel with this add a 10 uF to 47 uF electrolytic.