I was having major issues with rest mode about a month after launch; it was ridiculously unreliable (experienced all of the issues you described), and none of the solutions I read online worked. I went weeks, if not months, without using it again until I decided to try it without any games or apps running, which seemed to have better results. But it probably wasn't until Returnal that I felt comfortable using it fully again.
Not sure if one of the firmware updates ultimately did the trick, or if the games I was playing at the time simply weren't functioning properly with PS5's rest mode, but I've had zero issues since.