Why do we need to capitalize “I” and the days of the weeks in English?

No disagreement with the answers here. I’ll philosophise a bit more generally:

Each language authority or community ends up with a particular set of conventions about punctuation and capitalisation—or borrows them from a more prestigious language. You only become aware of alternate ways of doing things if you’re exposed to other communities. And it only becomes a problem if your language does not align to a single community. (This is a problem for languages like Esperanto or Ancient Greek, which is often capitalised and punctuated differently according to the country printing the text.)

The conventions about punctuation and capitalisation are memes, and they are only partly selected in response to functional needs.

So you can argue that I was capitalised because i looks odd (and ambiguous with Roman number 1); note that the older and dialectal variant ich (which shows up in King Lear) is not capitalised. And capitalising nouns does make some sense as a disambiguation, especially in a language with zero morphology like English.

But mostly it’s just a matter of fashion. It’s also an in-group marker: this person follows the community’s rules, so they are declaring allegiance to our community.

Fashion does change (English got rid of capitalised nouns); but conscious innovation by a single person does not usually get far.

As someone whose other language (Greek) borrowed its punctuation from French, I cannot tell you how much better English would be if it used French’s quotation dash. All those stupid “he said”s and “she said”s and “he expostulated” would just disappear: the dash would tell you all you need to know. Joyce thought the same thing, and he used quotation dashes.

And no one else in English does, because the quotation mark and “he said” convention has stuck.

And why has the convention stuck? Why don’t you get to be a mountain goat instead of a sheep?

Because language use does not usually reward the unconventional. People just scratch their heads and think you’re odd. Just like no one got fired for buying IBM, no one gets fired for following the language community’s rules.

You can violate the community’s rules to make a point. If you’re a poet, or an entertaining storyteller, you may be heard appreciatively. If the medium is part of your message—if you’re foregrounding the language, you can start your own memes, and be an agent in language change.

But much of the time, and particularly in formal writing, no one cares about how witty or innovative you are. They just want the content, and then they move on. So being a mountain goat in most contexts just makes you annoying. And wit and idiosyncrasy are a currency you should use sparingly in any case: the more persistently you do it, the less effective your deviations become.