Yet we see many if not most of the major efforts to achieve controlled
nuclear fusion as relatively uninterested in increasing density,
inertial confinement being the exception
I'm not sure you could call it "the exception" considering it's basically one of two major approaches to fusion and has received many billions of dollars in funding, second only to ITER.
The early papers on the topic, notably Nuckolls famous Nature article of 1972, suggested that implosion energies on the order of 10 kJ would be needed to reach breakeven conditions. KMS Fusion started getting neutrons the next year at ~1 kJ and so everyone was happy.
But then it simply refused to scale. All attempts to move from the 1 kJ to 10 kJ regime failed to improve the yields, and the 10k Shiva was a failure by any measure. From that point on we've been chasing one problem after another and we're still not there after another 45 years and 2.5 orders of magnitude more power.
The problem is that the plasma simply does not want to be compressed. You are compressing a light gas/liquid/frozen layer onto a near vacuum and that's basically a formula for R-T instabilities and shock-driven problems and all sorts of other issues. The result is that you get bits of the fuel driving into the center before others and the spherical compression you need to reach the required density is extremely difficult to actually achieve.
These issues can be overcome with additional compression, but the testing in Halite/Centurion suggested that 100 MJ of energy would be needed and that is well beyond anything we know how to build.
There are "middle road" approaches like MIF and MAGLIF that attempt to reach moderate density for longer times, but they have been beset by instabilities and have proven rather unimpressive in practice. Nevertheless, a number of private companies like General Fusion continue to predict practical fusion any day now using these methods.