Also keep in mind that , particularly for men, the nuclear family living was not the norm until the past few generations. Men used to spent most of their time in men/gentlement's spaces socializing/doing business while the wife was at home with kids. This 'loneliness' thing happens because those other spaces are gone now
Like in your solution you assume that people exist will be people who are young and have energy and have resources. These people do not exist. Not enough to take care of the massive number of old and sick people being created anyway.
Here's an illustration: https://ourworldindata.org/global-population-pyramid
This is referred to as an inverted population pyramid but really it ends up looking more like a bell. The world used to have lots of young people and not many old people. The flattening of the peak which has happened in only 30 years is stunning! This will keep getting worse until the end of the century.
Korea's population pyramid looks way worse than the one I linked above by the way. From age 60 down it actually is inverted. Again, there are no young people left to do the caretaking you envision. https://en.wikipedia.org/wiki/Demographics_of_South_Korea#/m...
Though I'm not really convinced that those opportunities for socializing no longer exist, I mean any medium sized city has meetup groups, pubs, and probably all sorts of other social outlets I'm not thinking of. The modern middle aged man's problem isn't meeting people, it's building relationships that have a closeness and intimacy equivalent to familial ones, which some people seem to think is easy but if it was these corpses wouldn't be rotting in apartments would they?
Big cities are filled with highly educated white collar serfs (sorry I mean professionals) who are great human resources.
In rural communities- the birth rate is way higher.
So what are you getting at?
It turns out that many animals, particularly mammals and including us, have evolved this behavior where they live with their blood relatives and take care of them, it has worked pretty well for millions of years.
Now it so happens that in many modern human societies there's a declining birth rate and consequently fewer families are being formed. South Korea's one of the worst off in this regard. You end up with a lot of people being alone and dying alone (the majority are male but women are far from immune).
Why are people not starting families? Well I haven't seen Korea-specific information but I think it's the same as most countries, people typically cite medical and financial reasons.
So basically you have to address cost of living, cost of raising a kid and rising infertility to address this, among other things.
However for a number of societies including Korea's it's probably too late. Koreans can't have more families and your solution probably won't work either. The reason is that the demographic collapse is so extreme, the population is increasingly too old to produce a new generation or take care of other people. There just aren't enough young people to go around. These societies either have to accept mass immigration or collapse. The next few decades will be interesting.