>The idea is that the secret key material is never compromised as it is assumed in all security cases.
That's not true, we have (perfect) forward secrecy, backwards secrecy and key rotation mechanisms because we often care what happened after the key is inevitably compromised. In this case the problem makes it hard to "rotate" the keys in a meaningful way, but I'm yet to see a proof it's impossible.
Even with n=1 you can get something useful. IIRC "on average" if you have ID x than the best population estimation is 2*x. Of course the error margin is immense, but it's still better than nothing.
It's good to consider this but... Plenty of sites expose user ID as a regular integer. In some cases you might want to avoid this (leaking user count to competitors etc), but I have never heard about anyone calling this a vulnerability.
That's not true, we have (perfect) forward secrecy, backwards secrecy and key rotation mechanisms because we often care what happened after the key is inevitably compromised. In this case the problem makes it hard to "rotate" the keys in a meaningful way, but I'm yet to see a proof it's impossible.