The problem is that there's no way for anyone else to determine a priori if your accusation is false or true. That's what the whole presumed innocent until proven guilty thing is about.
In reality, if you are accused AT ALL, you want your day in court and you want it to be fair. Even if you had committed a crime, if the police did something they're not supposed to that needs to get sussed out in court and you should go free.
Half the point of a trial is to make sure that nothing unfair is done by the investigators (police, prosecutor, etc). This is to keep their power in check so that they'll follow the rules. Otherwise it could get mighty tempting to fudge something a little bit "because we KNOW this is the guy!" and "we need to do the right thing."
Um, so? Then we need to allocate more resources to prosecute cases.
If I am falsely accused, I want my day in court. And I want it to be fair. The current system has problems on both fronts.