By simple definition, the unemployment rate is a figure that attempts to measure the percentage of the population that is not working, but would like to be, and, as such, is one indicator of how well the 'economy is functioning'.
Clearly, the groups that are excluded in the official number (but that are included in the U-6 figure), i.e. those who currently are neither working nor looking for work but indicate that they want and are available for a job and have looked for work sometime in the recent past (the so-called marginally attached workers), as well as those who are employed part-time for economic reasons and are available for full-time work but have had to settle for a part-time work (so called 'part-timers') fit the description of what the unemployment rate is trying to measure. (One could argue that 'part-timers' shouldn't be included, but that argument is not logically consistent with the aim of the employment rate: to see how well the economy is performing in meeting the desires of those in the population who want/need employment. Since there are those who desire full-time work, but can only find part-time work, by definition they should be included in the unemployment rate.) (God forbid we should mention all those 'trivial' issues of pay, work conditions, quality of work, benefits, self-fulfullment, alienating work, etc.)
Ipso facto (if my pretentious use of the latin term is correct), by simple first principles the U-6 unemployment rate IS the unemployment rate, not simply one variant thereof.