How to untar an archive without the root folder, but keeping the full hierarchy inside it?

Easy. Just use --strip-components=1 on the archive like this:

tar -xf archive.tar --strip-components=1

And as explained in the official tar man page:

--strip-components=NUMBER

strip NUMBER leading components from file names on extraction

With the logic being that if a path consists of foo/something.txt then the first “component” of that path would be foo/ so --strip-components=1 would effectively drop the foo/ from the path foo/something.txt.