The Swedish BankID has the same potential weak point. Any centralised system does.
The way TLS on the Web works is better: as long as the CA is up some time during the period I need to renew it is fine. Digital IDs should really work that way (probably with relatively short life spans just like let's encrypt: the digital ID could need to be renewed once a week for example, and it would opportunisticly renew when less than half the time is left).
Agreed, there should not be a tight (temporal) couple.
But it's a trade off. Long-lived TLS certificates have always had the cert revocation problem. OCSP stapling never took off, so in the end the consensus seems to have been to decrease expiry date. (Mostly fueled by Let's Encrypt / ACME).
Relying on expiration rather than explicit revocation of course also assumes (somewhat) accurately synchronized clocks which is never trivial in distributed systems. In practice it put's pressure on NTP, which itself is susceptible to all kinds of hairy security issue.
I like to think of the temporal aspect as a fail-open / fail-close balance. These centralized solutions favour the former, and that's why we see this resulting outage.
For anything as high stakes as eID you need real-time revocation checks, which brings you back to at least some level of centralization.
BankID is not government backed, and most governmental agencies have alternatives to BankID as well.
Italy's digital ID (SPID) works by having multiple trusted providers that can attest your identity. You can sign up with multiple of them, and if one is not available you could use another one. Not perfect (it's still centralized in the hand of 10-20 providers) but better than nothing. Unfortunately most people only ever signed up with one provider, and the government is now pushing for a more centralized digital ID istead (CieID).