It would be very very wrong to try to merge Job Centre with employment agency.
99% of Jobcenters in Germany are tagged as office=employment_agency, so currently the distinction you point out does not exist.
Should we tag the government ones with amenity=jobcentre in addition? But as Martin pointed out this might be a brand name in Germany and the UK.
Or should there be a subtag, like employment_agency(:tpye)=public, private? Looking at Wikipedia this seems like the better idea. In addition there is also the operator= tag.
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