Personally, I emulate anything that I play on a modern display and run that straight through HDMI, or DisplayPort depending on what the display accepts.

If I am playing something older, I usually have a working console for it, but if I don’t I still emulate it. If I have a working console, I plug it into one of my CRTs via composite (RCA) as that is the input my CRTs accept and the most common output on consoles of the time (like HDMI is today). If the game I am playing is on a console that I have that no longer functions, I emulate it and output via HDMI to an RCA adapter and plug that into one of my CRTs. This has some delay but it is not noticeable to me.

Sometimes I have fun playing modern games on a CRT in 480p with the same setup. The graphics look really good since I can max everything out with a good framerate, but the text and UI is basically unreadable, even with max UI scaling in most games. They just don’t make them with low resolutions in mind.