For touring on our LHT's...flat pedals and Salomon trail running shoes. The shoes have a stiffer midsole than road running shoes and this is plenty stiff enough for touring and designed for bipedal locomotion. Choose the outsole tread model that fits your needs. Gore-tex layer means no worries about covering shoes until REALLY heavy rains. Pedals can be cheapies or the most expensive out there, your choice. If touring where big bike shops are rare, pedal replacements are easily found.
I use and recommend clipless pedals on my road and mountain bikes but have never found an advantage when touring on the LHT. We tour on the LHT on all but the craziest single-track as well as dirt/gravel and pavement. If I were touring on a mountain bike, in the backcountry and looking for single-track, yes I'd be using a clipless system as well, but I'd also be carrying less gear and traveling lighter and searching out those single-track routes.
As always, YMMV but that's how we've done it for a long time.
Enjoy the ride,