I was just about to say, did you try switching to TwentySeventeen? Haven’t experienced those issues, so would be good to know. But switching to classic is not a bad thing.
The thing is that on a clean install all works just fine but on existing sites the issue above occurs. I have tried disabling extensions and tried it on Firefox.
Even on this site i get the above issue after updating. Any chance you still have the site package and can try it on local machine?
I have a site that uses Sections and new Blocks cannot be added. You can click the + and select a Block but nothing appears on the page.
On another site that does not use Sections, I am able to add Blocks as expected.