You do understand that besides thirds, fifths and sevenths, there really are seconds, fourths, sixths, ninths (same as second), elevenths (same as fourth) etc... as well, right? There even are intervals that are not named after a number e.g. the "tritone". The reason the 2nd and the 3rd note in a chord are called third and fifth is because usually chords are made with these intervals instead of dissonant intervals like seconds or fourths. It seems pretty clear you'd already know these things, so can you explain what's your issue with music note intervals?
It's that off-by-one nature of intervals that always bumps me.
The difference between note a and b is (a-b+1).
Calling an octave "an octave" feels to me like calling a numeric system with digits 0x0-0xf as "base 17"
By your logic, an octave/unison would be "seventh"/"zeroth"? (Note that "octave" literally means "eighth" in Latin.) I would think that could work too but the established terminology is not inconsistent. It's just 1-indexed. A lot of things are 1-indexed, in fact I think almost everything in spoken English is 1-indexed. A lot of mathematics, like number theory, is also 1-indexed. I think software engineers are a little too obsessed with 0-indexed things. I understand that things not being standard is annoying to us but pretending like this somehow makes music terminology broken is going too far.
I don't see how 0x0-0xf could be called base 17. 0xf is 15. Did you mean base 15? I think if mathematics terminology developed differently it could be called base 15. The same way binary is 0 and 1 but we call it base 2 because there are 2 digits, but we could totally call it base 1 too, who cares, it's all convention.
