I was running into the same issue as you were with that cryptic error message and really no details when you look at the log it references. I did have SSMS installed from before (went from pre 18 versions all the way to the current one).
The solution that I went with was copying the ISO to a local drive and running it from there, then it worked without any problems. I was worried about removing the SQL Server 2012 Native Client to figure out if that broke anything. It also works if you partially were able to upgrade certain parts of SQL Server (like the shared components) and not tried to roll it back to the prior version when encountering the error the first time.
Thankfully your post was there to help me out when my brain failed to make the connection with the SSMS issue (since I was already running 18.4).