However, websites can't control what browsers their users use, nor can they control how those browsers give feedback regarding load times.

They can control their own code.

Not saying that websites should implement this, but companies aren't going to want to point to browsers being the problem. Every company wants to (try to) control their users' experiences, for better or worse.

It has appeared to me that there are pageloading techniques that interfere with browser spinners. Many times a page is loading and there is a plain arrow where on other sites it shows a spinner.

absolutely, but we don't have a lot of control over that.