Quoted: A friend told me controlled feed actions are more accurate than push feeds.  Isn't that a myth?  Why are there so many precision rifles built off the Rem 700 action, a push feed design?   I read articles stating that the pre-64 Winchesters are more accurate than the later designs, but I think that's probably a production issue, not a design issue.  When Winchester reintroduced the claw in the mid-90's, I can't remember them making that claim either.

Accuracy is not affected by this at all.  Controlled feed is considered to be slightly more reliable than push feed.  This is only a concern when you are dealing with animals that are inclined to bite/claw/trample/maul you should a follow-up shot be necessary.  The difference is negligible.  

I guess a controlled feed action would prevent you from having the round fall out if you started doing cartwheels while you work the action. I think it'd be a pretty rare circumstance where the controlled feed (ie the more expensive Winchesters, right? anybody know of other examples?) would be an advantage over push feed (Remingtons-anybody know of other examples?). I suppose you could fall, be running or fall out of a tree stand and be dangling upside down from a safety harness and have some kind of dangerous animal after you, then the controlled feed would allow you to chamber a round while a push feed might allow the round to fall out of the rifle instead of being pushed into the chamber.

Since plenty of military and law enforcement snipers use rifles built off of Remington actions I suspect they are just fine.