If instead you used digits from –5 to 6, using arithmetic mod 12, it becomes obvious that e.g.:
-2 + -3 = -5
4 + 3 = -5
5 + 5 = -2
-5 + -1 = 6
4 + -3 = 1
etc.
Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc. Overall the simple heuristics are more obfuscatory than helpful IMO.
Where does it go wrong? Do those cases come up in practice?
Counting up and down the scale is a core use case for a notation for intervals. It absolutely needs to be well-supported. A 12-semitone approach is never going to match the usability of even the existing system.
I am obliged to point out that a 12-semitone approach is in fact part of the "existing system" (see: pitch-class set theory).
(Mind you, I of course think its usefulness is overrated, because I think the "atonal" repertory is tonal.)
Those interactions are pretty intuitive. Where defined, major + minor = perfect (considering an octave as perfect), perfect + major/minor = major/minor. As long as you remember which notes exist, you can't get it wrong, so you'll never get confused by a piece of arithmetic in an actual piece.