We've had 3 different Savage rifles that had problems feeding a cartridge into the chamber. The cartridge seems to "Pivot" a little bit (in the center of the case) in the magazine and this lets the base sag down to where the bolt won't pick it up unless you put your finger in and tilt the front of the case down which raises the base up to where the bolt will catch & feed it. Anybody else run across this and if so, HOW DID YOU FIX IT?? Godsdog

Experienced som problems with Savages, yes.
The problems have been the shape of the lips in the magazine. Reshape the lips, or take it to a gunsmith and make him fix it.

Just my opinion and experience, but the vast majority of feeding problems like you describe can be directly traced to problems with the mag box/follower/spring asembly...or ammunition with an OAL that is not compatible with the length of the box.

If a round is able to tilt fore and aft on its own, then something is preventing the spring and follower from exerting even upward force on it...or there is a hump on the bottom of the feed rails that doesn’t allow the full length of the body of the case to make contact...sort of a teeter-totter effect.