Industrial and organizational psychologists have researched this issue, with hundreds of studies spanning most different kinds of jobs in most developed countries now published. A work-sample test is especially strong for spotting people who need to do something right away after they are hired. For any worker who may need to learn new things on the job--and that is a lot of different kinds of workers, in a lot of different industries, and almost all managers--a "general mental ability" test adds to the predictive validity of selection applicants who will do well on the job over time as industry conditions and market pressures and technologies change.
Not necessarily. For example militiaries routinely recruit (hire) first, and then later figure out where to put the hiree. The same is less common but does happen in industry as well with entry-level college graduate positions, and in some cases even with higher-level positions.
Even if what you say is true, though, some companies intentionally hire people without the necessary skills to do a job, then train them. In that case there's no point in testing them for job-specific skills they don't have so what you want is a way to determine which candidates are likely to be successful after training.