The question is very broad; it could be split to literally thousands of questions of the type “why shouldn’t I use character ... in HTML documents?” This seems to be what the question is about—not really about code. And it’s about characters, seen as “weird” or “uncommon” or “special” from some perspective, not about character encodings. (None of the characters mentioned are encoded in ASCII. Some are encoded in ISO-8895-1. All are encoded in Unicode.)
The characters are used in HTML documents. There is no general reason against not using them, but loads of specific reasons why some specific characters might not be the best approach in a specific situation.
For example, the “little dots” you mention in your example (probably not dots at all but circles or bullets), when used as control elements as you describe, would mean poor usability and poor accessibility. Making them significantly larger would improve the situation, but this more or less proves that such text characters are not suitable for controls.
Screen readers could make sense of special characters if they used a database of various properties of characters. Well, they don’t, and they often fail to read properly even the most common special characters. Just reading the Unicode name of a character can be cryptic or outright misleading. The proper reading would generally depend on meaning and context.
The main issue, however, is that people do not generally recognize characters in the meanings that you would assign to them. How many people know what the circled plus symbol “⊕” stands for? Maybe 1 out of 1,000, optimistically thinking. It might be all right to use in on a page about advanced mathematics or physics, especially if the notation is defined there. But used in general text, it would be just… a weird character, and people would read different meanings into it, or just get puzzled.
So using special characters just because they look cool isn’t a good idea. Even when there is time and place for a special character, there are technical issues with them. How many fonts do you expect to contain “⊕”? How many of those fonts do you expect Joe Q. Public to have in his computer? In this specific case, you would find the font coverage reasonably good, but you would still have to analyze it and write a longish list of font names in your CSS code to cover most platforms. In the pile of poo case (♨), it would be unrealistic to expect most people to see anything but a symbol for unrepresentable character. Regarding the methods of finding out such things, check out my Guide to using special characters in HTML.