The circuit is an inverting integrator and a non-inverting integrator in a loop. That's how an analogue computer would solve a second order differential equation with no damping factor. In fact an analogue computer would use two inverting integrators and an inverting amplifier in a loop to do the same thing.
If you had two inverting integrators and an inverting amplifier in a loop, in theory the circuit should not oscillate unless given a 'kick' to start it off, like a pendulum. In theory it should then continue to oscillate indefinitely. In practice due to non-ideal operational amplifiers and capacitors, the oscillations could be expected to subside due to damping, again like a pendulum.
The non-inverting integrator in your circuit relies on close matching of the resistor and capacitor values to give good performance as an integrator. If the components are not perfectly matched you will get positive or negative damping. Slightly reducing the value of the resistor between pins 3 and 6 or slightly increasing the value of the resistor between pins 1 and 5 will increase 'negative' damping. You need a tiny amount of negative damping to keep the oscillations going indefinitely. Actually the two 1kΩ resistors come into play and provide some negative damping.
It was not shortening of lead lengths that got your circuit working. A few millimeters of wire at 1.5kHz isn't going to make any practical difference. What got your circuit working was swapping the components around when you rebuilt the non-inverting integrator. As the capacitors are likely to be less closely matched than the resistors, I guess if you swap over the two capacitors of the non-inverting integrator the circuit will stop oscillating.