I'd just say, strop as you see fit!
Apparently, according to Naomi-san and my barber, the linen is viewed as a "cleaner", it removes oil and cream, etc. from the edge before you hit the leather. And as for my barber stropping "very little", maybe 10-20 laps on leather. Just a few when he feels like it, apparently.
As for what's best for your razor, no one knows but you!