Even transfers through many cities need a test just to transit. You have to fulfil the requirements of your country of final destination, your country of transit and (in some cases) of the individual airlines.
In your case, for entry into the UK you need any type of test with negative result within 72 hours of your 1st international flight. For transit through AMS, KLM are correct.
in theory the person working needs a work permit (even if working and being paid abroad). In practice, if you're careful, who's going to know (don't broadcast it!).