This was a great read! You have excellent taste in music. For what it's worth, I think The Police had a reggae influence for a lot of their career--hence their album name "Reggatta de Blanc"! I agree, though, that most of their big hits didn't have much in the way of reggae undertones.
They did, and once again I didn't notice, until you pointed it out. 'Bring on the Night', 'Beds too big without you'.. both very Reggae.
Roxanne?
Yes!... I keep thinking Message in a Bottle but even 'Can't Stop Losing You', is reggae too. I never noticed it at the time, and still it needs pointing out to me to hear and realise it.