Liability waiver for construction road should be signed

I would not sign it either. You have a valid contract with the company. This must be fulfilled. It is understandable from the company's perspective. Perhaps they are saving on the necessary insurances. Additionally, as already mentioned, there is the extent of the liability exemption. No matter what happens in connection with the crane, you are liable. That cannot and must not be allowed.

Thanks already for the answers. I would now first inform the construction supervisor of the construction company that we are having the letter legally reviewed. We are also with [Bauherren-Schutzbund] and I hope to get a similar response now.

It's already a great start to the construction phase :-/ But as you already write, in the end, that's a free pass for all damages.

The expert from the Bauherren-Schutzbund at least said that evidence preservation should be carried out at the neighbor's basement. This is to ensure that no defects are transferred that were already present before. Due to the contract history (construction contract already signed and now this supplementary contract is coming), I should consult a lawyer regarding the extent to which the risk must be borne by whom and which specific contents and statements are acceptable in such a document.

Was the development situation known? Are there already obligations regarding the development in the construction contract? But the lawyer will clarify that.

I still have to take another look at the whole thing. The construction company will probably have included a paragraph like "every construction road is different and will require separate measures as well as liabilities." We now have an appointment on Monday with a lawyer from the [Bauherren-Schutzbund]. In addition, we will still carry out evidence preservation with the site supervisor from the [Bauherren-Schutzbund]. With a bit of luck, the current status and legal situation will be clarified next week and the delay of the groundworks will amount to only 2 weeks.

I myself have also taken out builder's liability insurance and construction all-risk insurance. I do not currently know in detail whether they would then cover it.

You should urgently ask the insurers about this (before any signing).