The decision is between 6H and looking for 7H.... but I'm
not sure how to look for it intelligently.
I think that's exactly right (as some others have indicated). Maybe 5NT
then 6H should show this kind of hand, but I don't think it does unless
you have some very detailed and unusual agreements. In practice 6S will
be taken as showing first round spade control (again unless you have
specific agreements), so I wouldn't bid that.
There's no reason to be terribly upset: you got to a great contract that
went down on a 6-0 break. Admittedly bad breaks are more likely after a
big preempt, but you can't anticipate 6-0.
I think West might well have pulled to 6NT with his S-Kx, but it depends
on how tricky or reliable he assesses the opponents as being. Psychic
Lightner doubles aren't common, but they can occur.