The "Critical Error with Start up and Cortana" is most definitely a Microsoft error and a Microsoft Problem, however.......
The S.M.A.R.T (Self Monitoring, Analysis and Reporting Technology) is a product of information produced by your physical hard disk indicating its' Status and in your case it is indicating that it is failing.
That has nothing at all to do with Microsoft or Windows or any other software product. It is a physical thing, solely related to your hard drive and is no more related to any Operating System than adding error in your car tires would account for engine noise coming from under your hood. You can add or subtract air in your tires but the engine is still going to be your root problem.
You hard drive could be empty, completely wiped and blank of all content..... It would still be physically failing.
AND
That is likely the cause of your Operating System problems not (as you seem to think) the other way around.
I'm sorry, and wish I had better news, but.....
You are just an unfortunate victim of a terrible and coincidental circumstance.
You were installing / upgrading your computer to a new operating system and had no way of knowing in advance that the hard drive was physically failing.
I can say, without any reservation, that with or without the Windows 10 upgrade, you would have seen that S.M.A.R.T error sooner or later anyway. The upgrade simply taxed your drive sooner and here we are.
