Thanks for that invaluable piece of information that changing your default web browser can allow you to also change your default search provider where Cortana is involved.
I honestly can't remember at this point, but I believe that Edge is actually set up as the default web browser under Windows 10 and you have to tweak settings to make IE the default. That makes me wonder if you could play the same trick with IE by changing it's default search engine? I don't use IE with any regularity, but instructions for changing its default search engine to Google on their help site read:
Internet Explorer 11
One can, of course, select something other than the Google search engine and follow the same process.