I should note that whatever you think of this explanation, it argues that unemployment is not an economic problem. It views the unemployed as happily and rationally choosing not to work, which thus suits their preferences better and thus is of higher economic value (neglecting, of course, the cost of providing unemployment insurance).

This seems to be a non sequitur:
If a specific unemployed person in a given state of the job market maximizes their utility by not taking any job offer, it doesn't imply that they are happy of the current state of the job market. They may be unwilling to work at the conditions that are currently offered, but they may be willing to work at different conditions.

The most obvious and extreme scenario is when you are starving and all the jobs available to you are physically strenuous and yet offer starvation wages: you'd probably better off conserving energy and trying something else, including begging or stealing (if you are caught, you will go to prison where the state will feed you).

More generally, in the standard one-shot prisoner's dilemma, you are probably going to defect, since it is the choice that maximizes your utility assuming that the other player also tries to maximize their utility. This doesn't imply that you are happy to play the standard one-shot prisoner's dilemma.

This means that if some intervention gradually reduces everyone's salaries by 50%, then most people will still be willing to work.

If all employers could pre-commit to do reduce everyone's salaries by 50% and stick to their decision, and it was legally possible for them to do so, and they could somehow survive the massive strike that would ensue, then employees would probably have to swallow it. But even if employers could legally unilaterally change salaries, they compete with each other to get workers, so they can't realistically do that.

It seems that you forgot to mention another obvious possible explanation:
Some people may just be unemployable.

Some people may be unemployable in general because they are mentally ill or physically ill (though they are still accounted for as "employable" in the statistics because of underdiagnosis).
The extreme example would occur if starting from tomorrow we counted chimps as people. All of them would be unemployed and permanently so because they would be always unfit for work.

Some other people may be employable under different economic/technological conditions, but not under the current ones. The extreme example would occur if we had robots with a IQ-160-human-level intelligence that could be rented for 1/100th of the current minimum wage. Who would be going to hire humans in this scenario?