The problem here is
not the notation. Mathematical notation is not perfect, and can sometimes be confusing. Let me say this in a minimally offensive manner, without being obfuscatory: the author of this piece does not understand the mathematics underpinning the notation he is using. The root cause seems to be the use of probability theory in a cookbook manner. It can hardly be surprising that confusion results.
The first example is a formula. An instance of magic, in the sense that you use it to compute, without knowing what it does. The $x_i$'s are not quantified. What are they? Are they real numbers? Matrices? Elements of some semi-group? How can you expect to understand the "formula" if the summand is not explained? At best, I can say that it is a formal sum of something. We can forget discussing convergence or it being well-defined. You can cook up arbitrarily 'nice' notation. It won't help. This notation is absolutely fine for someone who can infer that the support of the distribution of X is some denumerable set {x_i}, equipped with p.m.f. p.
A suitable definition of the expected value (as an operator) would have cleared up all the confusing with the variance and E[X^2] vs (E[X])^2. This confusion is not the notation's fault. It is the user's fault for not knowing what E[f(x)] means (for some appropriate meaning of the symbol f).
>> Only the first xixi is squared. p(xi)p(xi) isn't, because it doesn't make any sense in the first place. It should really be just PXiPXi or something, because it's a discrete value, not a function!
Functions are not algebraic expressions by which we associate one real number with another. In fact, we call p(x_i) the probability mass function. It seems to be a common flaw in many undergrad programs. Formulas and functions are never made distinct. The vast majority of functions f : R -> R do not admit an expression in a formula.
The example with the different notation for "derivatives" is a good non-example. The so-called Leibniz notation is used because it allows people to make statements with differential forms, without needing to invoke exterior algebra. If this is done correctly, statements such as "dy = f'(x)dx" can be made fully rigorous, if need be. Students are told that dy/dx is not a fraction, and yet it is used exactly as though it were. This confuses people - because they don't know what is going on. The dot-notation for derivatives is extremely useful in classical mechanics.
Notation is a clutch for succinct and meaningful writing amongst the initiated. One cannot expect to be able to use these tools without knowing what is going on, or by suspending a great deal of questions.
>> There must be other ways we can explain math without having to explain the extraordinarily dense, outdated notation that we use.
My final gripe with this post. We typically use clean and modern notation. It could be so much worse! Also, if we didn't re-use symbols, then we would run out, very quickly. Mathematics exists independently of the symbols we use to communicate it.