What about banks

Banks are centralized and can therefore unilaterally upgrade their cryptography. Bitcoin is decentralized and would rely on all users individually upgrading.

• Updating the signature scheme will need consensus in the sense that all nodes need to update after implementation of a quantum resistant signature scheme.
• Users of blockchain will personally need to move their funds from old addresses to new quantum resistant addresses. You won't need to move your bank funds.
• Lost addresses where people lost access to their funds will never be moved and stay vulnerable to quantum hacks. Blockchain doesn't know their users, can't communicate with them and won't be able to distinguish coins on lost addresses from coins from users who still have access but somehow have not migrated their coins after a quantum resistant update. So burning lost coins will be legally a big issue.

These are all issues specific for blockchain and not for banks or websites or any other centralized system.

Full answer: Banks and websites are centralized systems, they will face challenges, but decentralized systems like blockchain will face some extra challenges that won't apply for centralized systems.

Updating the signature scheme will need consensus in the sense that all nodes need to update after implementation of a quantum resistant signature scheme. Users of blockchain will personally need to move their funds from old addresses to new quantum resistant addresses. You won't need to move your bank funds. Lost addresses where people lost access to their funds will never be moved and stay vulnerable to quantum hacks. Blockchain doesn't know their users, can't communicate with them and won't be able to distinguish coins on lost addresses from coins from users who still have access but somehow have not migrated their coins after a quantum resistant update. So burning lost coins will be legally a big issue.

These are all issues specific for blockchain and not for banks or websites or any other centralized system.