I am definitely not confused because regardless of the fact that someone either has been breathalysed with what is called a "positive" outcome, i.e. the use of an official intoximeter at a Police Station, OR a person is accused by eyewitness accounts, i.e. signed written statements, those methods are not sufficient to convict them alone, that is always a matter for a court of law to decide.
In either case, what is being provided is some form of evidence which a court takes into consideration before reaching its conclusion, i.e. guilty or not. The reason I used the breathalyser example earlier was to highlight the fact that such evidence is far more indicative of alcohol levels in a person's system as opposed to a witness who has merely seen someone drinking alcohol, falling over, vomiting etc.
In the end analysis, however, neither is sufficient to prefer a guilty verdict which is something decided by a court which has as much evidence as possible in front of it which will enable the Magistrates / Judges+Juries to reach their conclusion(s).
Scotland probably has the best system because there is a verdict of "Not Proven" in cases where there is sufficient doubt of a defendant's guilt. England and Wales do not unfortunately have such a "tool" in their judicial armoury.