Thats far from reality. Just use the online form of BSI for disclosure. They contact the affected party for you. This way you optionally can stay anonymous and the vulnerabilities get fixed because BSI appears as the messenger.
They don't have to offer a version of the website without consent. As long as they inform you that the site will use cookies if you use it that should be GDPR-compliant.
They are talking about the chat template and not the system prompt. With current gen models, the system prompt is only part of a larger pretext that is passed to the model at the start of the "chat". The models are trained on a specific chat template with things like tool lists, reasoning budget, special feature flags and the "system prompt" formatted in a certain template.