The way it worked before was that you had basically a piece of paper with OTP codes and the website would prompt you for a very specific one.

How that would've prevented this issue: not at all. If the login service is down, having the piece of paper with OTP codes is worthless as the problem is not getting the codes (I can still get MitID codes with the OTP dongle) but the authentication website. The previous system was just as centralized.