We’d need to see the values set. I built it in Qucs-c and it oscillates (see the attachment). I removed the switch, since it serves no purpose in the simulation, while adding complexity to calculations. But perhaps you’re actually setting it to turn off power instantly after start? I also had to increase timespan, since this is a very slow oscillator. With the 60 second run time that meant 600,000 steps (100 µs) to calculate, which again is very excessive in general.
With the diode model I assume is used there,
(1) this circuit goes up and down between 14 mA and below 1 mA. Half of a cycle the current is too low for light to be noticeable.
The LTspice version misses the resistance introduced by the potentiometer. It’s not necessarily wrong. But it works as if the taper was on the extreme end, bypassing the entire length of the potentiometer. Not sure if that was the intention.
However, after stripping the circuit of irrelevant elements and rearranging it to a more common form, we get what the picture below shows. A triangle wave oscillator, but the Schmitt trigger feedback is wrong. At first I thought it’s your mistake in reverse engineering. But no, it seems you did it right. I’ll recheck later, when my brain resets. Just to make sure I don’t see things that aren’t there.

Assuming the reverse engineering is correct, I can explain it only in two ways. First is that somebody misplaced a resistor in design. But since it still does some oscillation (at least with LM358), it went unnoticed. Not the authors of this kit, as the design appears to be widely circulated way beyond this product. Second is that a long time ago somebody made that mistake, noticed it produces a more desirable effect, and shared it. But I can’t recall seeing that myself
(2) in the past. On the other hand I’d probably also not pay attention, until you asked now.
(1) 2 V @ 20 mA, noticeable conduction at a bit over 1.5 V, which matches many red LEDs. See the second image
this post for the charts of what diode “LED” models Qucs-c ships with.
(2) Which does not mean I did not. I might have seen that a dozen times, perhaps even soldered it myself. Brains are just very bad at memory.