Based on the facts you presented the brake issue is not a legal excuse, get a letter from your Insurance carrier stating they have taken care of the others persons damages, show it to the prosecutor and ask to change the charge to a Basic rules. That would at least keep it off your driving history.
You should consult an Attorney that handles Traffic cases in the jurisdiction you received the citation to evaluate your options.
In Georgia, unfortunately even traffic citations are criminal offenses. That being the case, however, you're presumed innocent until proven guilty beyond a reasonable doubt. Following Too Closely requires that the State prove that you weren't following the car in front of you reasonably and prudently - which sounds not like the case at all with your brake issue. A bench or jury trial may exonerate you without having to pay a fine or have anything go on your driving history. Show a prosecutor you aren't willing to back down and good things happen.
Ask the prosecutor to reduce your ticket to a non-reporting violation such as Too Fast for Conditions or Faulty Equipment. Otherwise, plead not guilty with a bench trial, go back to court, and if the other driver doesn't show up the case should be dismissed.
James L. Yeargan, Jr. is licensed to practice law in the State of Georgia. All information given is based only on Georgia law, and is not directly applicable to any other jurisdictions, states, or districts. Any answer given assumes the person who asked the question holds a Georgia Drivers License, and this license is not a commercial drivers license (CDL). This response, or any response, is not legal advice. This response, or any response, does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information. Any state specific concerns should be directed to an attorney who is licensed to practice law in that respective state.