If you have typical regular LEDs with typical limiting resistors, maybe 470R or 1k, powered with 12 V, then each will be drawing e.g. (12-2)V/(470 to 1k) ~ 10–20 mA current. Not an overwhelming amount for one LED, but if you have a bunch of LEDs and a marginal power supply it could be a bad combination.
Here’s some discussion of using so-called superbright, or as I prefer to think of them, super low current LEDs; they’ll give decent non-blinding brightness with very much smaller current draw:
For instance, red LEDs with 22k resistors: current ~ (12-2)V/(22k) ~ 0.5 mA.
In the Haillant LFO linked above there’s an LED powered by a (maximum) ±6.3 V triangle wave; with the op amp driver, the current is ~6.3V/(470) ~ 13 mA. The CE2 LFO has an LED powered by a variable amplitude, but presumably no more than ~ 5V, triangle wave, with a 10k resistor and no driver, so that’s around (5-2)V/10k ~ 0.3 mA. The LED seems reasonably bright in the video so probably it’s a superbright, but the description doesn’t say. Anyway, if it’s the LED current that’s the issue, then using those component values, it’s no surprise the CE2 has less effect on the voltage rail then the Haillant.