Right now it does. But not eventually. Quantum computer can reduce that to 80 bits.
Quantum computers are not magic tools that can solve any problem!
Then why do you think Satoshi invented Pay to Public Key hash? It's not just to save disc space on the blockchain.
P2PKH existed from early days, it wasn't invented later.
~ one day this person might wake up and realize all their bitcoins are gone.
That day we all wake up to an entirely different world where the banking system no longer works, military secrets are leaked and abused by the enemies and a large part of the internet would fall apart. You see the cryptography used in bitcoin and the security level is not just specific to bitcoin.
I can have 256 bits of security if I use a particular address type
Bitcoin keys provide 128-bit of security, you can't have more than that regardless of your address type.