Monterey 12.7.6 will not install any apps

Since I updated Monterey 12.7.6 on my 2016 Macbook I have been unable to install any apps via DMG. The process always hangs at the Validating packages stage, then issues an Installation Failed message.

I have tried all the recommended steps: giving the dmg Installer Full Disk Access in Security & Privacy preferences; disabling SIP in Recovery mode. Nothing works.

As of now, my Macbook is functionally useless with this OS version. Installing software is a fundamental requirement of any computer, yet Monterey 12.7.6 prevents this. Do I simply have no alternative but to upgrade to a more recent device?