Gotta correct you here. The push feed action will never give the reliability and positive extraction under any circumstances that a CRF with a claw extractor gives.
The difference may never show, or it may only show the one time your life is on the line.
You just don't have the surface area with other extractors that the big ole ugly claw extractor gives you.
The CRF will also feed reliably from any angle or position where push feeds are dependent on gravity. With the latter if you are at an odd angle left or right you can find yourself jammed at the very worst possible moment.
I own quite a few of both, mostly Remington 700's and Winchester Model 70's or customs built off of each and there definitely is an advantage to the CRF particularly when your life is on the line so when it comes to large/dangerous game I will always take the CRF over the push feed.