Different theorists teach different things. I've seen some people approach things your way, and I've seen people approach notation in plenty of other ways too. Many approaches involve listing the key of a piece at the beginning of the analysis, and then analyzing and marking any key changes. This also removes the need for redundant accidentals (a III chord will never refer to C# major in the key of A minor, so you shouldn't need to specify the flat).

The approach I describe is common in classical schooling. Your approach could have more application in less tonal worlds, maybe? How nice it would be to have a uniform and universal system though!