Things like dietpi-config and dietpi-software work, but this breaks apt, ls, dir, and other commands. I have to reformat my SD card and start with a fresh copy of the OS just to get basic functionality working again

Using the built-in SSH server to run commands and look at terminal output

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Usually it is not possible for apt package manager to break an entire system. Probably you have some file system corruption or a dying SD card. You could try checking for kernel error messages

dmesg -l err,crit,alert,emerg
This is btw not an error but an info that due to the missing package, possibly existing interactive package debconf options are not offered in a bunch before all package installs start, but during their installation. It is not required for anything than the comfort to pre-configure everything than go and drink a coffee, when you install a large number of packages .
However it shouldn’t break the system either of course.
The reason why your system breaks is that you install an ARMv7-only PHP on an ARMv6-only RPi. I’m also wondering why it breaks the whole system instead of only leading to a failing PHP service, but it’s known like that. I already reported this to Ondrej years ago, but since it is explicitly an ARMv7 repository, he didn’t find it reasonable to have a closer look.
If I remember right, it really is impossible now to do anything, especially purging the package again, right? So I’m AFRAID you need to reflash and stay with PHP7.4.
… ah another thought and explanation: The package pulls a lot of very basic system library upgrades from that repo, like libSSL. So every command with relies on any of those libraries is doomed to fail. Makes all sense then.