Hello, this is a forum where even laypeople can write! On the topic of construction supervision in connection with KFW funding, I say nothing but refer to the KfW.
On the topic of construction supervision: I do not know the situation. Who is the site manager? Who employs them? What you usually need is someone who has knowledge of as many trades as possible, is only obligated to you, and (!) has a lot of time to be physically and mentally present on the construction site at the right moment very often. That’s why I would take this one ("bauvorlageberechtigter Bauingenieur: 2000€ für 7 begehungen incl. Prüfung Unterlagen"). If needed, when a defect is identified but not recognized or remedied, I would then occasionally bring in an expert specialized in the relevant trade (if the "construction supervisor" is not already competent themselves).
Regarding liability. I suspect they protect themselves to ensure that if a defect is present but overlooked (which can also happen during supervision, after all, he only wants to visit the site 7 times), no liability arises. I can understand that. And in my opinion, it shouldn’t matter to you, because your contractual partner (architect, craftsman, BT/GU/GÜ) is liable, who is usually responsible for the defect.