I'm a pure mathematician, so take what I say with a grain of salt. My sense is that "constant nomadic life" is not a particularly accurate view of things. I think it is true that the number of years of postdoc people often do is slowly increasing (or maybe more accurately, the sort of research track-record that's required is increasing, and it takes most people more years of postdoc to achieve that record), but I mean that taking 4 or 5 years after Ph.D. to get a permanent job is quite common, rather than 2 or 3. Those 4 or 5 years might mean two multi-year postdocs, or a 3 year postdoc and a year or two at institutes like Max Planck or MSRI; after that time, it will get hard to get temporary jobs. I think a fair number of people get through 1 or 2 postdocs and then decide to get a non-academic job, but I suspect most do not have reasonable options for continuing in academia in temporary positions (sometimes they have permanent options they decide are not worth it).
If you look at the job listings on mathjobs, there are currently about 500 TT jobs, 170 postdocs (which will generally not want to hire someone after more than 1 postdoc), and 120 "non-tenure-track" positions which is a big grab bag of things, mostly an equal mix of mislabeled postdocs and teaching positions (many of which are continuing). If you figure TT jobs are hiring for 20-30 years, and the postdocs generally for 1-3, the disparity isn't THAT bad.
As an editorial note, I will mention that I think too much funding has gone into postdocs at the expense of TT jobs in the past few years. I know it sounds really good to help the next generation of scientists train, etc. it would also help them a lot if they could just settle down and stop moving. However, I think things have not gotten quite as dystopian as you suggest.