In those days though subtle variation in MHz actually made quite a noticeable difference in how the system performed, even noticeable to an average user - the differences MS/Sony are talking about have an impact but is more than subtle enough that few people can really even tell them apart.
HD vs 4K is almost never going to be noticeable for TV viewing: you have to be within 8 feet for a 65" screen, closer still for 55" or 42". However, for games it's different. People often sit closer for gaming than they do for TV viewing (filthy console peasants - they sit on the floor with their livestock), and text looks better in 4K.
You are quite right screen size, screen resolution and screen viewing distance are all integrated. Even if you are a Golden Haired PC gamer :-) you still have to consider the three "S" guideline when selecting a monitor. A good rule of thumb and this also applies to TV viewing. "If you can see the pixels you are sitting too close, either get you eyesight checked or if you don't have eye issues then you need a smaller screen or a higher resolution screen."
So getting 4K right is probably within the realm of what humans can distinguish. 8K OTOH - you basically have to be so close that the whole TV won't fit in your field of view (so maybe there's a point in 8K movie theater projectors, but not home use).
Actually, its allot more complex than that and I am definitely not going to give a lecture on it since there are many good sites that can do just that. Sometimes the best solution for a person who prefers gaming to watching a TV show or movie is to get a monitor for their console rather than argue with household members as to what they want to use the TV for.
With a monitor, you treat the console as if it is a PC with the added convenience of having your PC connected as well. Obviously, you do need a monitor with one or more HDMI ports so you can connect your console to it and an appropriate PC port. I actually use my monitor's second HDMI port and can toggle between PC and console in seconds.
Now comes the fun part of choosing your monitor and again rather than me rant there are plenty of web sites that can help and some that will confuse as well. Don't believe all the hype you should get what is best for you by being aware that screen size and minimum comfortable viewing distance (see Note) will dictate your screen resolution. Of course, there are other factors to consider as well such as refresh rate (in Hertz and the higher the better), HDR for 4K screens (you need HDR10 for PS4 and XB1) and latency which can be affected by HDR. Another rule is "don't buy cheap". It is sometimes well worth paying that little bit extra for quality but you can also waste money of features you don't need although monitors are not like TV's.
Note: When I say "minimum comfortable viewing distance" I am talking about the distance your face is from your screen. In the case of a PC (particularly when doing PC tasks) that may be from 30cm (1ft) to double that although for a console game you may wish to sit further back. This should be taken into account when selecting your screen size and eventual resolution.