That's completely hypothetical. I could say "cat" and "dog" should in the future both be called "dog." That doesn't really excuse calling cats dogs.
Perhaps a new standard of spelling could be created and, as you suggest, "it's" and "its" could be replaced by "its." But that's not what you're doing when you write "it's" instead of "its;" instead, you are writing a word that is different from the one you mean.
"Its" has a meaning now, today, that whoever parses you language will use. Perhaps you will convince everyone to overload that meaning with a new one, but in the meantime write what you mean.
In the meantime, I'm dyslexic, and you don't seem to be very considerate of that.
Additionally, what I said is not hypothetical. It's a linguistic fact. Your comparison of phonetic spelling to replacing "cat" and "dog" with one word is a linguistic falsehood.
Perhaps someday falsehoods will be as accepted as facts, but in the meantime, speak the truth.
Writing "pier" instead of "peer" is, just like writing "cat" instead of "dog," wiring something different from what you mean. Your reason for confusing "pier" and "peer" doesn't make them the same. Perhaps it's hard to get the right answer, but that doesn't make a wrong answer right.
As for your dyslexia, I don't think it's relevant. Your case for phonetic spelling wasn't specific to dyslexics.
If "pier" and "peer" were merged into one word, then there would be sentences that are ambiguous. No such ambiguities arise with "its" vs "it's". They aren't even the same parts of speech.
You also have an incorrect notion of right and wrong. Read up on some linguistics and get back to me. E.g., there is nothing more "right" about "Standard Written English" than there is about the dialects spoken in inner cities. Both are equally good, full rich languages. Though certainly mistakes can be made within a dialect, you must understand, in order to be a civilized human, that for many native English speakers, their native language is not Standard Written English, but rather a regional dialect. Consequently, when they are in a situation where they are expected to communicate in Standard Written English, they are being expected to communicate in a language that is not their native language, and yet many people will not afford them the allowances that we make for people whom no dialect of English is their native language.
As for my dyslexia, it certainly is relevant, as this entire thread, from when I started posting on it, which is a direct ancestor of this post, and yours, is about whether it is moral to deny people jobs just because they make such an unimportant mistake. It's a mistake that never actually matters, except to the pedantic, and it's a mistake that dyslexics are particularly prone to. Not because they don't understand the difference, but rather because they often can't see when such mistakes are made.
Perhaps a new standard of spelling could be created and, as you suggest, "it's" and "its" could be replaced by "its." But that's not what you're doing when you write "it's" instead of "its;" instead, you are writing a word that is different from the one you mean.
"Its" has a meaning now, today, that whoever parses you language will use. Perhaps you will convince everyone to overload that meaning with a new one, but in the meantime write what you mean.