I am very familiar too with the Police 'jump the traffic' as they join. They want to observe and see who is paying attention and who are not. So when they join, like you describe they can make enough progress that the traffic coming up to them is less aware of their presence and so are acting 'normally' and then the Trafpol can judge who reacts to them and who doesn't etc etc !

I saw one police car (oncoming) with full B&T's coming up to a massive line of traffic in the M6 Nth roadworks (about Preston area but I have travelled a long way today !) I just hoped they were paying attention and helped him / her get through quickly ...
Another I saw also oncoming was in clear road and the light traffic did not look like he was getting any one in his way.
How anyone can miss the B&T's is frankly hard to comprehend.
I can however understand that people can 'freeze' as too little reminds and clear messages are provided to the public.