Unable to restore OS

Given the symptoms as a whole it appears that Mac's startup disk has failed.

A "DOS-type" window is a mystery to me though. If you installed Windows using Boot Camp it might be an explanation for its appearance, but that doesn't change the fact the disk has most likely failed regardless of what Disk Utility reports.

You might try erasing the startup disk according to Use Disk Utility to erase an Intel-based Mac – Apple Support (UK). After that, reinstall macOS (Step 10) and migrate that Mac's contents from its Time Machine backup. The startup disk might continue to work, perhaps for a long time.

<<. "The volume OS X Base System appears to be OK". >>

That is great to know, but MacOS Base System typically of size about 2 GB is the stripped-down system used for Recovery and Installer, only. MacOS Base system is NOT the active system and may not be on ANY drive (it could be a ROM-resident disk image).

The default name for the active MacOS Volume might be Macintosh HD or similar names, or it can be changed to something of your choosing, but NOT MacOS Base System.

More telling is that if there was no other Volume offered, your drive may have died, as John Galt has already suggested.

You have to be prepared to face the possibility that your drive has died. That is not ALWAYS the case, but is likely.

Drives sometimes have a few spare blocks that can be substituted for ones that go bad. But you can't know for sure until you have erased the drive and have re-written the drive and see if it can hold new data.

NB>> YOU need to erase the drive and test it, if desired. A technician can NOT take the gamble that the drive MIGHT come clean (it might not and they wasted all afternoon on it.) If you ask a technician to check the drive, it will ALWAYS be replaced, because that always works.

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When Magnetic drum and Magnetic disk devices were first invented, the recording surfaces were VERY expensive to make, and the prospect of one tiny defect sending the entire recording surface to the scrap heap was just too onerous. So they invented a way to 'work around' small localized defects, provided they were small. The concept is called 'spare blocks', and is still in use today, although not in exactly the same way.

Originally, the Drive controller reserved and set aside a few blocks in each recording 'neighborhood' that were never allocated or counted in the total capacity. when a Bad Block was detected in the factory, the controller would substitute a spare block from the pool of spares and make a note in its internal tables that whenever you mentioned that bad block number, the controller was to fetch that data from the spare block instead. later a separate lookup table was added to use for blocks that went Bad 'in the field'.

SSD drives use a variation called Over-Provisioning, to effect the same sort of results. So there is a slight chance a small number of Bad Blocks might not be a death sentence for the drive.

Hi,

Try to resetting NVRAM, then startup in Recovery Mode could help your issue.

Reset NVRAM on your Mac - Apple Support

Use macOS Recovery on an Intel-based Mac – Apple Support (AU)

The:

Macintosh HD - Data (that your User volume, Not the easier-to-recover-from System Volume)

Input/output error

Sounds generic, but is actually VERY SPECIFIC.

Some of the blocks in the directory have gone Bad. not just bad data, but Bad Blocks. The data they were supposed to contain is unknowable, and the Directory is damaged, and therefore the drive is us-usable until re-written with new data.

¿what is the date of your most recent backup, and by what method?

🙁

I think both the Apple Store and the drive recovery service are being overly optimistic. SSD data tends to be unrecoverable.

The Input/output error is a definitive indication that Mac's startup disk has failed.

... will restoring the OS overwrite my personal data?

No; reinstalling macOS is nondestructive on Macs. But as Grant Bennet-Alder wrote the Input/output error implies that at least some of the data stored on the startup disk is already gone. That log entry will not occur unless the low-level drivers responsible for disk I/O tried and failed to read or write data, multiple times.

macOS incorporates a backup system: Back up your files with Time Machine on Mac - Apple Support. You just have to connect a disk and turn it on.

To get your Mac fixed start here: Official Apple Support. Hard disk drives are easily and inexpensively replaced.