Winchester claims to make left handed rifles these days, not sure as to which models/calibers. Actual availability could be a problem, depending on local outlets in your location, as well as caliber/models
Other left handed makes are Savage, Remington, Weatherby and maybe Mauser. Also, there are some "custom" makers claiming recreations of Modrel 98 actions and Magnum Mauser actions, and I've seen mention of left hand as an option. They are expensive, I would think.
As to the so-called "controlled feed" as opposed to push feed, I shot Model 70's in National Match Course competition for years, mine were all "push feed", never had a problem with any, both 30-06 and 308. I also shot left handed, which with IRON SIGHTS, presented no problem at all. Of course, the Model 70's I had were Target Model, about 10 to 11 pounds, which I guess would not be called "light weight rifles". They were pleasant to shoot however, especially in a 1000 point aggregate match, that's 100 shots for record plus sighters.