HP Easy Scan - "HP Scanner 3" will damage your computer
This message happens when the certificate used to sign software gets revoked. The OS does check for revocations online so yes it blocks apps even if you don’t update the apps or the OS.
For HP printers and scanners, the problem is the revocation of the certificates HP used to sign their software, not that macOS checks for revoked certificates. Things are working technically how they should, but from a support and culture perspective, I am certain thousands (perhaps millions) of people are realizing “suddenly” that this is built into their OS.

https://eclecticlight.co/2020/10/23/why-have-my-hp-printers-stopped-working-how-to-check-their-software-signature/

The vast majority of HP apps for macOS (which are slightly different than the driver stubs needed for printing) were signed by now revoked certificates.
You are correct in scanning software will not work unless you ignore revoked signed apps or update your software. Here is the package we pushed to thousands of Macs around noon yesterday to address yesterday’s specific certificate revocation.

https://ftp.hp.com/pub/softlib/software12/HP_Quick_Start/osx/Installations/Essentials/hp-printer-essentials-S-5_14_8_4.pkg

Run software updates and you should see updates for HP or worst case see if you can get support from them (they will be very busy the next few days while this all sorts out)

If you want to check your software, here are two apps I love and use to check certificates and signing status.

https://objective-see.com/products/whatsyoursign.html
https://eclecticlight.co/32-bitcheck-archichect/