How do modern displays output data to pixels?
Do new displays still write pixels one at a time?


If we talk about a video interface to a display, then yes, pixels are still "drawn" one at a time on the screen. 

If we talk about the CPU drawing pixels there are various methods. The CPU can write 4 bytes (32 bit processor) or 8 bytes (64 bit processor ) at a time into memory. 

But most display memories are not written by the CPU whilst the contents is displayed. 
Instead the memory is updated and then as a whole passed to the video controller. This is often done during the vertical blanking period. This method is preferred as it give a more 'steady' image. (Also see "bit blitter").


  I saw a video showing slo-mo footage of a display, in which it was clear that all of the pixels are on at the same time.


Just because you update/change one pixel (write it) at a time, that does not mean it will be off when it is not written.  Think of a large amount (~2 million) of three-colours* LEDs. You change the colour of them one at a time, but in between they still keep giving out light of the last value written. (We are no longer using an electron beam to write one spot on the screen)

*In reality there is grid of 3 LEDS. See wikipedia


  However, this seems like a very costly method considering the massive number of pixels in modern displays


"costly" is a relative term, but yes you need some memory to store a screen image. 1920x1080 is a common format as is 24 bit colours these days so you need 1920x1080x3=6.2Mbytes. Taking into account what I wrote above, you need twice that memory for a steady picture. 

Most CPU's have 4G of memory so if we set aside a part of the for the video you use ~0.3% for your display. But if you have a graphics card then it will have it's own video memory.