Japan also has an extremely high population density, and there are not really alternatives for most. It's not really a fair comparison to systems that have to handle urban sprawl and car cultures.
I'd be curious to see if different lines/regions (in Japan) are profitable or not.
Whatever is holding back your trains, it's not lack of density. Here are the population densities of some places:
Japan: 327/km^2
England: 434/km^2
US Acela Corridor (Boston-NYC-Philadelphia-Baltimore-DC plus minor cities and outlying areas): 931/km^2 ... nearly 3x Japan!
Yet Japan's rail network is superior in every way to England's, which is in turn superior in every way to the Acela corridor. When I google "acela frequency", the first result is "12 kV 25 Hz AC", which really says it all, doesn't it?
Then there's Switzerland: 207/km^2 and much tougher terrain, yet their train network is Japan-tier. Compare that to California, with 250/km^2 and no serious rail connectivity at all, despite Sacramento-SF-San Jose-Los Angeles-San Diego sitting practically on a straight line. There are only 16 trains per day between San Fransisco and Los Angeles, and it takes over 10 hours. These are two of the most important cities on Earth, it's ridiculous! There should be at least 10 times as many trains, and 1/3rd the journey time.
Look at Texas: Dallas, Fort Worth, Waco, Austin, Houston, San Antonio; jointly they are 20 million people and all exist in a triangle less than 500km to a side. The region is perfect for high-speed rail. There should be hundreds of Shinkansens zooming back and forth across the desert each day. In the manner of Texans, they should be even bigger and even faster. But they don't exist.
The US as a whole is thinly populated, but it still has several extremely wealthy and dynamic mega-regions that have more than enough population density to support world-class high speed rail networks (plural).
It's weird that you chose Japan, which includes lots of rural places, including the island of Hokkaido, but you chose "England", instead of the United Kingdom. It would be much more informative to select the area inside M25, then overlay in central Tokyo. It won't even be close. I guess central Tokyo is 2x the density of central London.
Fair. I picked England because transport in Scotland and Wales are devolved matters. A better comparison in Japan might be the island of Honshu, which is only slightly denser than England (447 vs 434). So I think my overall conclusion stands, that density isn't that related to rail network quality. That extra 13/km^2 isn't why Honshu is full of bullet trains and England isn't.
No, I think it's unfair. If you look at a map of the shinkansen lines in Japan, they're not just in Honshu; they span the entire length of the main islands (not including Okinawa, of course). They go all the way from Sapporo in Hokkaido down to Nagasaki in Kyushu, and beyond. The one to Sapporo goes through a deep tunnel under the strait.
So a more fair comparison would be: why doesn't the UK have bullet trains stretching from Cornwall to northern Scotland (past Edinbugh), and another branch going to Belfast in Northern Ireland?
But yeah, your point is exactly right. Density has nothing to do with why Japan has the rail network it has. It's entirely because of political will. If it were just about density, you'd only see shinkansen lines connecting the 3 major metro areas: Tokyo, Nagoya, and Kansai (Osaka/Kyoto), since they're the largest three metro areas, and they're all relatively close together. Instead, Japan is happy to build a shinkansen line even out to Kanazawa, which only has 500k inhabitants (but is a popular tourist spot), in addition to cities at the ends of the island chain (Fukuoka and Sapporo).
This probably includes suburban and rural areas. If you look on Google maps sky view, you will see the western half of Tokyo-to province is mostly rural (forest, mountains, and farms). The central part is made up of 23 "ku's" (cities). According to Wiki, the density is about 16,000/km.
I'd be curious to see if different lines/regions (in Japan) are profitable or not.