[__d]

I’m not an expert here, but …

CRTs needed to be refreshed to keep the phosphors glowing. But all screens are now digital: why is there a refresh rate at all?

Can’t we memory-map the actual hardware bits behind each pixel and just draw directly (using PCIe or whatever)?

[kbolino]

I think you're assuming that LCDs all have framebuffers, but this is not the case. A basic/cheap LCD does not store the state of its pixels anywhere. It electrically refreshes them as the signal comes in, much like a CRT. The pixels are blocking light instead of emitting it, but they will still fade out if left unrefreshed for long. So, the simple answer is, you can't get direct access to something when it doesn't even exist in the first place.