This couldn't be further from the truth. Even just in special relativity, if you want to compute the age of an object on a very fast ship, you need to use the L

This couldn't be further from the truth. Even just in special relativity, if you want to compute the age of an object on a very fast ship, you need to use the Lorrentz transforms, which categorically use a metric for the space-time. It's at the base of everything else in relativity.