Restore from Time Machine stuck

That's a good decision.

After (successfully) restoring a Time Machine backup a Mac simply presents the usual login screen.

Given the previous description of your difficulties I am beginning to suspect that iMac's hard disk drive is no good.

If you decide the restore operation will never finish then force the Mac to shut down as you propose.

All the problems you describe including those in your other Discussion continue to implicate a failed startup drive.

Sure, but Apple won't replace anything if their tests indicate it's OK. I prefer to evaluate a hard disk's suitability for continued use by booting from an external device.

But what happens to all my apps my apps that I have on the hard drive?!

That's one of those questions you have to ask yourself, because the answer depends on the degree of inconvenience you would face by completely erasing it (which you really must do to definitively rule out the possibility of problems with it). What works for one user isn't necessarily best for anyone else, but options will fall into one or more of the following general guidelines:

• Completely restore a Time Machine backup, which erases the target volume first. If one backup doesn't work for whatever reason, choose another, which is the reason one and only one backup isn't sufficient. A successful Time Machine system restoration results in the usual, mundane login screen as though nothing ever happened... just your system, as it existed, at that point in time.
• Erase the target system and reinstall macOS. Then, migrate individual User Account information from a Time Machine backup (once again, the ability to choose from any number of TM backups is ideal). You can choose to include or exclude Applications and other broad categories to migrate.
• Erase the target system and reinstall macOS, then selectively install essential apps from original sources, followed by restoring work products (documents, files, folders, photos, movies, music etc) individually. "Original sources" will vary according to how those apps were obtained. The App Store is easy because you can just download apps again. Downloads from "elsewhere" (developer sources) are another story, but again retaining original .dmg files (for example) from a Time Machine backup makes it easy. They might require license information or installation keys or similar types of authentication so it's incumbent upon you to retain that information along with those products.

You might notice that I haven't included offsite backups, which is an option but I don't use any of the products you mentioned. All of them have advantages and disadvantages, and the best way to evaluate their suitability for your own particular unique needs is to simulate a disaster: suppose one day you turn on your Mac and it just won't light up. Or suppose your iMac is just gone. What do you do? Use caution with commercial off-site backup services like "iDrive" because some of them will not back up macOS or programs, citing licensing concerns and other excuses. Study their terms and conditions, and confirm their marketing hyperbole does not conflict with reality. Better yet, conduct that emergency drill and see for yourself.

I wasn't aware any backup dates were manually removed, which is certain to corrupt a TM backup and hinder its usefulness at best. If nothing else it is almost certain to render a complete system restoration impossible... as seems to be the case here. Selectively restoring individual items is likely to be unaffected though.

I understand problems existed before you did that, but it just compounds the uncertainty.

In my opinion there are no hard disk "scan" utilities worth anything, although if Disk Utility indicates a failure it is likely to exist. If a disk is faulty it can be used to failure but should no longer be relied upon. That's one of several reasons one and only one backup is not sufficient. Taken to a logical conclusion no spinning hard disk should be relied upon. Everyone's needs are different, but if you ever find yourself in a position where the prospect of losing any one particular hard disk at any given time causes the slightest concern, then you have to reevaluate your backup strategy.

What I really think is that no one knows.

It seems that way.

The only thing you haven't done is to install macOS on an external drive though. Until you do that the status of the internal one is still undetermined.

A successful outcome might also convince Apple something isn't right with it. You should go to them being able explain how you tried A, you tried B, etc... now what?

When you erased the internal drive and reinstalled macOS, did it work OK?

It’s done just restart the computer.