I've got both an old TV (with SCART RGB connection) and a PC CRT monitor.
Initially, I tried the standard S-Video TV-Out on graphic cards. Quality was decent, but it's restricted to 480i/576i. I was looking for something better with more flexibility, and bought a Scart-VGA adapter (UMSA).
Generally worked out as I intended, downside seems to be that 60 Hz modes trigger NTSC and not PAL60. At least that's how I would think explains the color differences, especially reds become oversaturated.
The Laptop that I used for the VGA connection unfortunately broke down, so I'm relying more on the common emulation devices Wii and the first Xbox again. Not quite as flexible and powerful as a PC, but it's enough for most of the standard stuff I'm currently playing, like Genesis and SNES games without special chips.
The PC monitor is smaller than the TV, but I was amazed that it's possible to also display 240p resolution modes with some fiddling and the right graphics hardware. If you turn down sharpness a bit, it looks very good imho.
As for LCD's, most of the old games I've seen look rubbish whether on TV, tablet or phone without shaders and/or the right filter combinations. But on a powerful enough device with the right settings, it can be looking good. I'm not sure I'd call it a realistic mimicking of CRT technology yet, using my own TV as reference it seems a bit off still. From what I've read and seen, there can be quite the differences between CRT's themselves though, e.g. Sony Trinitons have supposedly relatively strong scanlines.