Website can't be read by Google

Hi there,
my client’s website can be reached by url without a problem, but it seems to be impossible to be read by Google (Page speed, ads, etc.) or any other automated online services (Linkedin ads, SEO online tools, etc.). There’s always a HTTP 404 status code answer in those cases.

It doesn’t seem to be a general server response issue because I put a test.html in the root directory that can be read by Google.

I deactivated all plugins, but unfortunately that didn’t change anything.

There’s no robots.txt in the root directory and the search engines are not excluded in WordPress.

As I didn’t create this website originally and was only called to fix the issue I suspect that one of the plugins wrote something in the .htaccess or the header.php!?

Do you have any idea what the problem might be or how I could fix it without rebuilding the whole thing? I mirrored the website to a private workspace but the problem still remained.

Thanks!
Philipp