Boot failure

Boot recovery (or internet recovery, and with most current version, using Option-⌘-R ) and try to install macOS or restore your backup onto an available (empty, can be erased) storage device, and boot that.

This to test whether the internal storage is failing (in isolation), an external hard disk allows operations (with degraded I/O performance), and can verify that the rest of the problematic Mac is working.

Run diagnostics or hardware test. See if that identifies any issues.

Boot recovery (or internet recovery, and with most current version, using Option-⌘-R ) and try to install macOS or restore your backup onto an available (empty, can be erased) storage device, and boot that. This’ll tell if your internal storage has failed, or if there’s some other issue.

Hard drives are one of the most typical failures here, though many other sorts of failures are possible.

So to confirm, you have re-installed macOS on an external storage device, and then booted from that external storage? Was that then stable? I'm here not suggesting reloading onto nor booting from your internal storage, I'm suggesting a means to use external storage to test whether the internal storage is failing.

Something here has very clearly failed, it very clearly involves the internal storage, and the question then becomes whether it's the internal storage device, or the I/O hardware that connects to storage, or some other core part of the Mac that has failed, is unclear.

To differentiate a failure isolated to the internal storage device from a failure elsewhere in the Mac, installing on and booting from an external storage device can be a useful test. And it can allow continued operations, while repairs are organized. If the failure is not isolated to the internal storage device, then repair costs increase, and external storage may not or will not work reliably.

I’m focusing on the hard disk here because a failure of the hard disk can manifest as reported here including sometimes working and sometimes not working, and because it’s also one of the few components here that can–given available external storage, which is unfortunately not available here–be bypassed and tested separately, and because external storage can be repurposed for backups and other uses.

You’re headed for a repair or replacement, yes.

If you’re running hardware test, this Mac is fairly old.

So next, try an external install and external boot.