You could get some token from the website. It could include encrypted service name and policies, like rate limit, that the authority should enforce. The client passes the token to the eId authority. The authority signs it and adds timestamp, but no user info. Client gives token to the service. Something like that. This is a bad top of mind example.
I think we'll need to rely a lot more on eID in the future. I think it can be done in a good way but then it needs to be thought through before it gets adopted. And we have to be able to trust the eId institutes.