You only need to brief cases for the first few weeks of law school or for however long it takes for you to be able to recognize how to read a case, dissect it into its components (procedural history, facts, issue, rule, reasoning), and take from it whatever it is you are looking for. Once you are able to do that STOP BRIEFING!!! Get a case brief supplement or look up the cases on lexis or westlaw or whatever and call it a day.
Absolutely right. Case briefing was designed in a simpler age for a simpler purpose: to put all the law student ducklings in a row behind the mama prof duck. (Who was then the papa prof duck, come to think of it, although I'm not sure papa ducks spend all that much time worrying about whether their ducklings are in a row, behind them or elsewhere.)
In law school, cases exist for one purpose and one purpose only: to highlight a point of law. It could be a rule. An exception. An exception to an exception.
So, focus first and primarily on that point of law. USE the case, sure. Know the case and know why it's there. But don't spend more than five minutes scanning it. Then and only then do you read it--and only the parts that are relevant to the point of law. Absolutely don't spend more than five seconds on procedural history, etc., unless that is the point of law at issue. (If, for example, it's in Civ Pro.)
Reverse the process: know WHY you're reading the case before you do. In law school it's easy to think of cases as the donut. Nope. They're a hole, or more specifically a black hole, into which an entire semester can be sucked--with absolutely no benefit. Focus instead on what is important: learning the rules of law (just one or two per class, usually) and on how to apply those rules to new facts.
If you get called on and don't know the procedural history, etc.--and I'm trying to think of any time this ever happened outside of The Paper Chase--just say you don't know. Don't worry about it. It WON'T affect your grade (as long as you're not obnoxious about it). Don't worry about being embarrassed. Really. Who cares? Focus on the rule of law, instead.