Note: italics such as "2nd" refer to my proposed notation; existing notation is written without italics: "2nd".
Why is the interval between C and D called a 2nd, as they are 1 note apart? Surely this should be called a 1st? Similarly, the interval between C and E should be a 2nd, from C to F should be a 3rd, and so on; rather than an "octave" we have rather a 7th. ("heptave"?) Naturally, the interval from middle C to middle C is a 0th, as the same note is at a distance of 0 from itself.
This would greatly simplify matters. For instance, if one makes a triad by stacking two 2nd's, then the resulting interval between the top and bottom notes is a 4th, as one would expect from 2 + 2 = 4. The same would be true for any addition of intervals. In contrast, in the currently-used notation if one stacks two 3rds, the resultant interval is a 5th. This is clearly the sort of off-by-one error that results from one-indexing. Are there any reasons besides existing tradition (which is of course quite a compelling argument) why this notation persists?