The folder icon with question mark that you're seeing appears when no bootable media is found, and in your case seems indicative of SATA cable failure or hard drive failure. Your local Apple Store can help diagnose this (they have a more thorough in-house hardware test OS that isn't publicly available), but I'd place my bets on cable failure, given your symptoms and how failure-prone they are known to be.
If it's just the cable, they can replace it for you (typically £100 for parts and labour), or you can get the part and tools for around £15 and replace it yourself if you have the time and patience. If it's the hard drive itself, you'll obviously need to replace it, and your data is likely lost — in this situation, don't attempt to recover the data yourself; you'll do more harm than good. Instead, consult a data recovery specialist if you desperately need the data and don't have a backup.
To attempt some better diagnostics at home, make a bootable Ubuntu USB and see if that detects your drive. I've had some luck in the past with that, in situations where a SATA cable has failed sufficiently that the Mac EFI and macOS (including Recovery) refuse to show it, but Linux will do its best to provide info. When in Ubuntu, open the Disks app to see if your drive is detected. If it is, check the SMART report (select the disk in the left sidebar, click the three-line icon in the upper-right, click "SMART Data & Self-Tests...") If all is reported good, the cable is almost certainly the issue.
P.S. "obviously I can’t repair a disk that doesn’t appear in [Disk Utility]" — Likewise, you cannot use
fsck to check a filesystem that the OS cannot see, hence why it says everything is fine.