A file system is a small index at the front of the drive and a set of tables scattered across it: the partition table, the boot sector, the master file table on NTFS or the catalogue on HFS+ or the object map on APFS. The files themselves are elsewhere, and they are the last thing to be touched. When the index cannot be read, the computer has nothing to show you and, in the only way it knows, offers to write a fresh one. That is the format prompt. It is the computer's shrug, not its verdict.
Two things make an index unreadable. The first is damage to the index itself on a drive that is perfectly well: a disconnect while writing, a crash, a power cut, a bridge that quietly translated sectors differently from the last one. Those are rebuilt on an image, and the odds are good. The second is a drive that cannot read the sectors the index lives on because its heads are weak or the surface has bad sectors there. The prompt looks identical; the drive is not, and it must be imaged head by head before anyone goes looking for the index.
That is why the bench never tries to repair a file system on the original drive, and why chkdsk and First Aid are dangerous here. Both write. On a damaged index they can make a bad situation permanent by rebuilding it wrongly; on a weak drive they hammer the weakest part of it. The drive is imaged first, once, and the file system is put back together on the image where nothing can be lost by trying.