I'm sorry you don't find it great (I still do). Integers are not characters.
Integers are numbers like -1337, 0, and 42.
Characters are things that compose strings of text.
These are not the same kind of thing at all. Just because APIs may be leaky, and some of these APIs are held in very high regard doesn't change that fact.
In the end, integers, floating point numbers, "text\n", emojis etc. are just sequences of bytes. You choose to acknowledge it and take advantage of it, or you don't.
A bit late, but bytes are very often the smallest addressable unit. That's why they are "special".
Which answers your second question: "bitstreams" would be terrible because they are not well connected with a hardware reality. Unless you have bitstream-oriented CPU, it is a bad idea for a basic type to go against the hardware.
Why even have types... Well, yes, there are languages without type checking where the notion still exists. For instance Forth has no type checking but two types are implied: the "byte" type and the "machine word size" type, maybe three if you count strings.
All you've convinced me of is that the + operator isn't defined for characters. Which makes sense. Trying to tell me that something called char is really just a byte in disguise (while true in some popular languages) is just irritating and misleading to me.