FWIW, the US military considers push-feed to be perfect for hunting the Taliban and other baddies.
And I would venture a guess that those rifles get a thorough work out in some of the worst conditions imaginable.
All true but missing the point as bears are shot at close range, rather than sniped at long range, and usually done with bolt actions. The "push feed" rifles being used by our military for similar CQB use are typically able to operate on full auto.
Still, a well tuned PF bolt action being operated by a competent user will work just fine for brown/grizzly hunting