My suspicion is it provides more visual indication that a new sentence is starting, but it bothers me because I think a period / exclamation / question mark seems sufficient to me. Plus, if a proper noun begins the sentence, then it's not as obvious it's a proper noun. Are there any problems if the convention of capitalizing the first letter of each sentence were eliminated?
You don't have to. It's just convention.
if i taip in all lower case and space and spel sum things funny thne its harder to read
But if I spell, capitalize, and punctuate the way you expect, it's a lot easier to read. Why? Because as a literate reader of English, you've practiced reading text that follows convention to the point where it's automatic. When you see taip, you stumble over it because it's unfamiliar; you haven't practiced reading it nearly as much as type. And the same thing is true of sentence boundaries that aren't marked by capitalization.
but of course, it's not a requirement. you can still tell where sentences begin and end as long as i use punctuation. the only time capitalization really makes a difference is when you capitalize (or fail to capitalize) a CAPITONYM--and even then, it's likely that you'll be understood. in fact, in some languages capital letters aren't used at all, so clearly it's possible to be understood without them. you could do away with them entirely, if you wanted.
But all the same, you should probably follow convention as a courtesy to the reader, unless you have a reason to do otherwise.
Personally, I do not think that there will be a huge impact if the convention on capitalizing the first word in a sentence were eliminated. I also agree with you that it exists maybe because there used to be a visual necessity to do so. However, if we imagine how Latin was used, without capitalization, and worse punctuations, I guess capitalization does not provide any harm.