Getting 240p to display properly on HDTVs is a huge pain for retro gaming enthusiasts.
It largely comes down to the quality of the scaling hardware within the display and the assumptions it makes about the signal. I knocked together an RGB-to-component converter for the Apple IIGS recently and tried it out with the LCD displays I had on hand: three TVs (two name-brand and one not-so-name-brand) and a monitor that also has component input (and S-video and composite, in addition to the usual VGA and DVI). The monitor kinda worked, but it chopped off the first line of text IIRC. The not-so-name-brand TV didn't work at all. The other two TVs worked: the entire screen area was visible. Color quality and 40-column text were pretty good. 80-column text was usable, if a bit fuzzy. I had hoped to use it with the monitor in the computer room, but the missing line of text would be a bit of a problem (it's like it's not syncing up until it's too late). None of them are as clear as the ancient NEC MultiSync 3D I normally use with it, but who knows how long that will continue to work? It already takes several minutes to settle down and run right after a cold start. I suspect a CRT TV with component input would be better than the LCDs, but I haven't had one of those for several years.
(While the adapter is intended to plug straight into the IIGS's RGB output, you could lash up an adapter to use it with other devices. In addition to red, green, blue, and composite sync, it also needs +12V and -5V. It only cost me about $50 to build, and maybe $20 of that was two extra boards from OSH Park, which ships in multiples of 3.)