As someone who spent 7yrs flying Budgies in that part of the world, (only ILS on 17 at ABZ in those days, VOR at INV,Wick and Kirkwall, SRA at LSI) I think that the correct decision was made by the skipper.
He set off to the destination with an expectation of getting there, makes an approach as the reported vis. is above limits but isn't able to land.
Sumburgh is his preferred commercial alternate (same land mass) and an attempt is made there.
To commence an approach the reported vis. would have led to an expectation of landing.
Unable to land and diverts to his weather alternate ABZ and arrives with above minimum fuel but forsees that he may land with less than minimum fuel and is obliged to declare an emergency.
I made several similar decisions in my time at ABZ and can remember asking for an expeditious return to Heathrow on a JSY (bigger aircraft) after setting off with max fuel (landing weight JSY restricted) when the weather changed somewhat abruptly during the approach - Bournemouth was closed due to a blocked runway in case you ask!!
Yes, it would be nice to carry an unlimited fuel supply and worry about all the possible failures (why stop at just a flapless landing add an engine out and total brake failure) but then no aircraft could fly anywhere.
Was the operation safe and the answer has to be yes.