You did not give the police "written testimony", you gave them a statement. Testimony is given under oath or affirmed as true under penalty of perjury, a statement is just a statement.
The statement is not part of the "proof". Proof is an evidentiary burden that must be met by the evidence to obtain a result. If this is really a criminal matter, the police must prove that you committed the offence "beyond reasonable doubt". However, driving offences are almost always civil penalties, not criminal ones, so the burden of proof is the much lower "on the balance of probabilities".
The police are under no obligation to enter your statement into evidence. They might, but if they don't and you want the judge to know about it, you have to enter it into evidence through your testimony. The police are there to prove you guilty, they have very little incentive to help you prove your innocence.
A trial works by the prosecution presenting their case, calling witnesses who testify, can be cross-examined by the defense and then re-examined by the prosecution. Then the defense has their turn in much the same way. The defense doesn't have to mount an active defense because the burden of proof lies with the prosecution, however, if the only evidence is that given by the prosecution, that burden may not be too difficult to reach. That said, for traffic matters in local courts, this will not necessarily be done with the same formality as the Supreme Court but whatever latitude is allowed is up to the judge.
When a witness testifies they are not normally allowed to refer to written materials - they are supposed to do it from memory. Courts will usually allow some witnesses to refer to written materials: police to their notebooks and experts to their reports but this is the exception rather than the rule. For civil cases, the parties can agree that testimony will be by written statement that are provided before the trial - this is done for reasons of efficiency, however, this is not usually the case when someone is being prosecuted.
Since I have nothing to add to my story, am I allowed to remain silent and to read the written testimony instead?
Your statement may or may not be in evidence at the trial. If you want it to be, you should testify that you made a statement to the police and that you are happy for that to be your testimony - you will probably be allowed to swear as to its veracity and the prosecutor can object to some or all of it on the normal grounds (relevance, hearsay etc.) for which the judge will make a ruling, perhaps striking out the offending parts.
You can choose not to testify, the court will then make a decision on the evidence advanced by the prosecution.
Assuming they both have the same content, could a judge put more trust in the oral testimony than in the written one?
There is no written testimony q.v.
Could remaining silent compromise my defense?
Yes. If the evidence consists of a police officer saying you were speeding (say) and that's all, the judge can choose to believe the police officer.
Could the judge ask for clarifications (or other questions) on the written testimony?