Is "I" a proper noun? Why is it always capitalized when "he" and "she" are not?
'I' is not a proper noun, and the reasoning behind the rule for capitalizing it is likely for aesthetics.
As you suspect, 'I' is no more a proper noun than 'he' or 'she.' Unlike those words, however, a lowercase 'I' simply doesn't look good on its own, and may be less legible than its uppercase counterpart. There isn't a complete consensus as to why the switch was made, but as 'I' developed from the German 'ich,' people wrote their single 'i's larger than in other words, and eventually switched to the capitalized version.