Do I reinstall the whole system, install Pika, point it at its old backup folder and have it restore? If so, what does it actually restore? Does it originally back up apps, their data and whatever I have in my home folder, then it restores all of that to the new system? Or does it only back my config files and home folder?
Pika backup is made for backing up files, rather than a full system. By default it only includes your home folder, and excludes things like cache folders. Per their Github: "Pika Backup is designed to save your personal data and does not support complete system recovery."
The file restoration feature is more like plugging in a USB drive with files stored on it. Pika backup mounts your backup as if it were an external drive, then allows you to copy your files from it.
You might want to consider a system backup tool like Timeshift instead. That seems like a closer match for what you're describing.