The strobing is due to the way they're powered. When converting AC power to DC power, you can either convert the positive half of the AC sine wave (half rectified) or both positive and negative sides of the AC wave (full rectified). Cheap lights use half rectifiers so as the AC feeds in, the light is only getting powered half the time and off the other half of the time. This happens so rapidly that we don't really see it with our eyes, but with a camera it's very noticeable. AC cycles at 60 cycles per second (in the US), so it makes sense that you're seeing it at 120 shutter speed as this equates to 60fps.

I'd consider talking to some of these venues about it as I assume they're typically used for events that'll be filmed, so using shitty lighting is ruining it for everyone same as if they had the toilets sitting in the middle of the dance floor.

So LEDs are generally very bright by default, and there's a limit to how much you can really dim that. The solution used is often to flicker the LED at high speed, imperceptible to human eyes, called Pulse Width Modulation (PWM). Basically, by changing the percentage of time the LED is on (the width of the ON pulse) you change the perceived brightness off the light.

Cheap LED designs do this slower, cause the hardware and LEDs are cheaper. It's not really to save energy, but to adjust brightness and manage heat.

Also, probably more importantly, cheap LED bulb designs just don't deal with AC current as well, so you get the 60Hz flicker from the electrical line cause that doesn't get regulated out correctly when converting to DC.

So what I heard a long time ago is that all LED lights essentially strobe on and off constantly, faster than the human eye can detect, though i’m willing to bet the better ones are constructed in a way that you wouldn’t notice at all (whether with the naked eye, or through a high shutter speed camera, as you mentioned), but the shittier ones strobe more frequently and not at a rate that’s as smooth/consistent as the better ones. As a little anecdote, I bought a lower quality light-tablet for tracing a while back, and was getting crazy headaches after using it for maybe an hour or so. Had to stop using it, and I think it must be an example of what poor quality LEDs are like.

Not sure why, but the high power LEDs you see in cars do this if they're cheap or done poorly. Mine came with like an ant-flicker ballast or something.

You're seeing the 60Hz sinusoid caused by using AC electricity. 60 peaks, 60 troughs -- but the like is actually turning on and off 120 times per second when on AC, unless it's first converted to DC. Cheap LEDs just feed AC.