Until OP provides better information, it is impossible to provide truly useful information. Because there are many people who have rented cars from many companies in Switzerland and may have had related experiences, knowing the specific name of the company involved is critical. This has nothing to do with "pointing fingers" it has to do with either wanting or not wanting useful information.
As a general matter:
1. You are responsible for damage to the vehicle during the rental period. Whether you caused it, noticed it, or did anything negligent, is irrelevant (although that may affect your ability to recover from someone else).
2. Unless you noted the damage on the appropriate form at pickup, it can be presumed to have occurred on your rental. Not to berate you, but you have already recognized this.
3. Depending on the rental contract --- hence the need for the name of the company --- it may be that LDW or any other ancillary services are voided if you did not comply with local law which may have required a report as soon as you became aware of the damage. Thus, if such a provision is in the contract, it became your duty to report as soon as you were advised by the rental company.
The advice to report this now may well be solid, but is all dependent on what is in your contract and a good deal of what you need to know may come from others on this board if you provide the details.
Don't get into an email battle with the company. It is useless. Focus on whether there is any way to determine that this was pre-existing damage, whether the EUR 7,800 is the reasonable repair cost (seems a bit high even if the entire door is replaced), and the terms of the LDW.