I've had issues with rest mode since the first week - and don't use external storage - and even recently did a full factory reset about a month ago and still have issues - so might need to reset again, but this time using a usbstick to reinstall the firmware from PlayStation, or it is possibly a problem with my old launch PS3 registered PSN account.
My PS5 still kills the bluetooth/usb - so won't work with pad, and still doesn't work with usb connection - if left in rest with a game running for more than a day or so and brought out of rest by power button or hdmi-cec - so the machine still responds through hdmi-cec but won't let me access restart or shutdown , just rest mode via tv off. And because I then have to hold down the power button for 30secs to power off, the PS5 on restart thinks it was a user poeroff error, so never lets me report the real error.
I've now started using rest mode for only charging the pad, so I now exit the game before using it, but even then, the other day CoD had an issue doing a background update it failed to complete in rest-mode .. well I think it did, because that was the error notification in the UI after reboot - when the system came back up after a forced power button shutdown, after the system locked up straight after selecting a profile coming out of rest mode. I then went and checked if CoD had a update to download - seeing as it claimed it didn't complete - but oddly, it was fully up-to-date.
edit:
My PS5 is also connected via UPS - which was bought to protect the TV from my kids turning of the plug for the extension when they were really little.