No, actually. I haven't solved the problem. All I've done is to put nothing in the General Settings for the Site Title and Tagline. I've set the title tag by using WordPress SEO plugin, but I think that's kind of a stopgap measure.
The checkbox you describe doesn't exist on the site--which is strange, I know. The problem started when my client updated the theme, which of course wiped out all of the customizations that had been made to the site. I was able to restore the customizations, but then this problem with the inability to choose to not display the site title and tagline cropped up.
If you can give any advice on how to fix this, I would appreciate it.