While some public transport issues increases with density (harder to e.g. dig new tunnels), one of the major issues with public transport at scale is the cost of laying rails and digging tunnels, and amortising those costs over larger populations makes it far more affordable. Once you have the rails, density means you can offer far more frequent service, which makes a huge difference to users. Being able to just assume there'll always be a train within a minute or two is a huge deal to get people to actually use public transport. Density also works the other way and reduces the need in many cases, as well as drop delivery costs and give new options (e.g. I can get stuff from one of the local Amazon competitors couriered over in 2 hours) that first become cost effective when density is high enough to create big enough markets.
> (though there are artists' renditions, most of them dystopian)
That may be so, but density to an extent also creates opportunity. E.g. my local road takes up roughly 45,000 square metres. Of this, roughly 1/3 is gardens where each family in effect only gets use of a very small portion, leaving many of them largely unused. Another roughly 1/3 is pavement, front yards mostly used for bins etc., and the road. Near our local station, new highrises are going up that will house almost the same number of people on a couple of thousand square metres of space. That's a lot of space freed up.
In other words: The question is how far you want to go. If you slash space wasted on roads and wasted duplication, and increase heights, you can get massive increases in density almost everywhere and still end up with more desirable and practically usable space for most people to actually use for things other than transport that is only necessary due to the low density. If you go further than that, and take away all the green space etc., then yes, you get a dystopia.
But you don't need to go very far. E.g. where I live is relatively average density for London. Increase the density of the London metro area five times, and you could empty the entirety of the rest of the UK into London. If every road like mine was replaced with high rises, you'd be able to achieve that with ease and still be left with vast empty tracts of land for new parks (and allotments for those who actually use their gardens today). London makes up well below 1% of the land area of the UK.
I'm not suggesting it'd be a good idea to create a single megacity for the entire UK population. But achieving the density would free up huge amounts of land even if we set aside space for lots of big new parks. It'd also save vast amounts of energy used on transport by cutting distances for most travel.