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I hate PWM car taillights, but I doubt that LED screens will have the same problems. PWM is not the only way to control LED brightness and even PWM can be OK if the frequency is high enough. Also, projectors already flicker.

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What other methods are there to control LED brightness?

PWM is just one (very inexpensive) way to implement a DAC (digital to analog converter). Other methods actually set the output voltage to the target voltage, such as a R-2R ladder. I don't think R-2R ladders are used very much in practice anymore but I can't remember any other implementations off the top of my head.

The other main way would be using a constant current driver[1]. By adjusting the amount of current to the LED, you can control the brightness.

[1] https://electronics.stackexchange.com/questions/256336/does-...

LEDs are current devices; any mechanism for controlling current precisely can be used to control LED brightness. there are a lot of them.

running a smaller, continuous, current through them.

PWM is done because it's cheap to implement given a fixed power rail and a digital control : need a single/couple of power switches that can even be integrated on die for low-ish power targets. continuous things usually need some kind of discrete filters, that means more components etc

Eizo has screens that use DC dimming:

http://www.eizoglobal.com/library/basics/eyestrain/index.htm...

I never tried them though.

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