And this can happen with less than a few percent friction cost?
Explaining why this will fail, let alone describing how it did indeed fail where it was tried is econ 101 material. As someone else pointed out, look up Lump of Labour fallacy.
1. Not all jobs are created equal, and not all unemployed persons are qualified or capable of performing all jobs. There would still be a mismatch in positions available and workers to fill those positions; without "perfect" alignment a community productivity deficit would continue, moving the costs around instead of eliminating them.
2. It'd become politically desirable to disproportionately point these efforts at industries where it'd be easier to integrate the unemployed ranks (lower skill set barriers to entry). This would have an effect of reducing profit margins in those sectors given the increased overhead of a larger workforce for an unpredictable amount of productivity change. One expects if the total labor hours are the same, economic output of an individual entity in that economy would be equal while personnel costs would increase.
3. There will always be a small portion of the working populace out of work for legitimate reasons.
And, please do a little Googling on seasonal work. In the US, for example, millions of jobs fluctuate between seasons depending on buying and production trends. That's not small peas, so to speak.
I suspect the only reason most people accept it is that they learned it at school so it must be true.