In my setup the website is inside the build folder. The content of the build folder is what I upload to my server, which is why I put .htacesss there.

In your case, you have another level. Try with this .htaccess (inside the public_html folder):

FallbackResource ./build/index.html

FallbackResource should point to your index.html file, so you need the right path based on where your .htacess is.

I hope everything is clear now