Most CRT-based television sets sold in the US will accept any combination of horizontal sweep rate that's within a few percent of 15.75kHz and vertical sweep rate that's within a few percent of 60Hz, with any arbitrary phase relationship, even though the FCC has for many decades required that broadcast stations output signals that are within a tiny fraction of a percent of 15,734.27Hz and 59.940Hz, respectively, with a precisely-specified phase relationship. Electronically, it was easier to make a set with a timing reference that might deviate from the ideal by 5%, but then make the set accept anything within 8% of that, than it would be to design a set which would know or care if the input signal was within 1% of the specified ideal.
Many if not most home computers and video games in the 1980s generated video with horizontal and vertical frequencies that were a percent or two away from the ideal frequencies and lacked the FCC-mandated phase relationship, since doing so was often easier than matching the specifications precisely, and because for various technical reasons doing so could improve the appearance of the computer's output on television sets of the era.
If the person who designs a digital television set wants to support 1980s computers, it would not be hard to make the unit give a better picture than a 1980s television. Many television sets, however, make simplifying assumptions about the incoming video signal and will produce an inferior picture (or in some cases no picture at all) if those assumptions don't hold.
Getting the best flat-screen picture from vintage electronics requires using a television set whose designer was interested in supporting such devices. Unfortunately, I've never seen support for such equipment regarded as a marketing feature. Some sets work well and some work poorly, but I've not found any way to predict whether a particular set will work with a particular device other than by trying it out and seeing what happens.