Site TItle suppression

I have a custom header with my site name in it so I don’t need a site name that shows up on my site anywhere.
If I don’t enter a site name in my blog, the browser tab shows blank and I’d like it to say the name of the site. However, if I enter a name into the site name, the browser tabs show up correctly but then the site name shows up on top of my customized header.

There is a checkbox that says “Display Header text” and if it’s checked the site shows up in white, if It’s unchecked it shows up in gray like a shadow.

Is there a way to fix this so it doesn’t show on the site, but does display the name in a browser tab?