The importance of controlled round feeding is directly related to the intended use of the rifle.
The Remington extraction & ejection system is simply not as strong or as reliable as a Mauser system with large claw extractor & mechanical ejector.
The push-feed system can bobble a round on feeding, under certain circumstances.
For deer hunting, the Remington (just to use it as a common example) works fine. For most non-dangerous game a push-feed is perfectly viable.
But, consider this- If you're within 50 yards of a charging bear, rhino, lion, etc., PF vs CRF does become an issue.
With a PF action in a typical hunting scenario, if you miss the shot because the action didn't cycle (on the second shot, assuming the first didn't connect) on either extraction or feeding, you just lose the meat. If you're being charged and the action lets you down, the meat you lose will be your own.
That's the difference & the reason why most people who understand it use a CRF rifle in situations where subsequent shots may literally mean their own life or death.
And, you can't cite police & military use of the Remington, widespread though it is, as justification for saying PF vs CRF doesn't matter.
Think about it- Police & military users are snipers. Snipers are long range. Snipers use cover. Snipers are stationary. Snipers HIDE! Snipers are not in immediate danger of being charged, by either man or beast. Snipers typically have time on their side in setting up a shot. Snipers place their primary emphasis on the first shot (where PF & CRF are equal), whereas those up close place a roughly equal importance on follow-up shot capability. A sniper also has time on his side, at least regarding personal danger, in getting off a second or subsequent shot if necessary.
The two uses are apples & oranges.
If you need to have a dependable rifle for maximum bolt-action reliability under a wide range of conditions that include relatively close threats from dangerous game, the CRF is widely recognized as the better way to go.
I've had a Remington ejector plunger freeze up solid in the bolt, the rifle would not eject brass, the gun had to be either turned upside down to drop it out or the empty brass had to be picked out with thumb & finger. If I had some big claws coming after me, that's unacceptable. If it involved just watching a buck run away laughing, it'd merely be annoying.
Like any other firearm decision, use determines choice.
Denis