Why are proofs not written as collections of logic symbols but are instead written in sentences?

You have not translated the pages from Apostol's book into mathematical logic. What you have done is to transcribe them into your own idiosyncratic shorthand, which may be useful to you but is less than meaningless to anyone else.

Let's start with the use of the symbol $\stackrel{\mathrm{def}}=.$ In normal mathematics, this tells us that the notation on the left is defined to represent the expression on the right in a general way. For example, when we write

$$ \cosh x \stackrel{\mathrm{def}}= \frac{e^x + e^{-x}}{2}, \tag1$$

it is a definition of the $\cosh$ function. In a definition of this sort, a symbol such as $x$ is a variable that can be substituted, so Definition $(1)$, above, tells us not only how to interpret $\cosh x$; it also says how to interpret $\cosh y,$ $\cosh t,$ $\cosh a,$ or $\cosh b.$ For example, Definition $(1)$ informs us that $$ \cosh b = \frac{e^b + e^{-b}}{2}.$$

In your notes, you start with the definition $$ [a, b] \stackrel{\mathrm{def}}= \text{closed interval in $x$-axis}. $$

Now, setting aside the fact that there are four English words on the right-hand side of that definition (what were you saying about using symbols rather than English text?), you have just defined a bracket notation for us, "[" followed by a variable followed by "." followed by another variable followed by "]" and you have informed us that this is a closed interval on the $x$-axis. Now it seems strange that your variable names do not occur on the right-hand side of this definition, and in fact this does make the definition relatively useless in strict logic: which closed interval is denoted by $[a,b]$? But worse still, on the next line we find out that changing the variable names changes the definition to a closed interval on the $y$-axis, not the $x$-axis.

If you actually succeeded in translating the pages to pure logic, along the way you would realize that the labels "$x$-axis" and "$y$-axis" are hints to help you visualize things, not part of the strict logic of the mathematics itself. You really need only define the closed-interval notation once.

I would say that some of your uses of $\stackrel{\mathrm{def}}=$ are actually logical definitions of symbols and notation. But many are not.

If you have a good definition of the product of two sets, it is not necessary to write out your interpretation of $P_x \times P_y$ as a "definition." It would already be defined and (logically) unnecessary to write. By the, way, symbols such as "$\ldots$" do not belong to the notation of mathematical logic; they are (again) merely hints to understanding.

You also seem to tend to use "$=$" to signify "is a" rather than the standard symmetric, transitive, and reflexive notion of equality. For example: $$ Ƃ:Q \to \mathbb R = \mathrm{SF} $$ would mean the same thing as $$ \mathrm{SF} = Ƃ:Q \to \mathbb R $$ if you were writing in the language of mathematical logic; and the meaning of the line in which it appears would still be ambiguous. (Is SF a mathematical constant like $\pi$?) If you actually were writing in mathematical logic you might have defined SF as a predicate, written in the form $$ \mathrm{SF}(Ƃ:Q \to \mathbb R). $$

Later on that same line, however, you write $Ƃ:Q_{ij} \to \mathbb R,$ contradicting what you wrote earlier. The domain of $Ƃ$ could be either $Q$ or $Q_{ij},$ but it cannot be both in the same definition. It seems you want to say that the restriction of $Ƃ$ to $Q_{ij}$ is a constant function, but you have neither the logical notation to describe a restriction of a function to a subdomain nor to say that a function is constant. You end up defining $©_{ij}$ as a synonym for $Ƃ:Q_{ij} \to \mathbb R$ but not saying anything about new what the function does.

Frankly, without using Apostol's text as a Rosetta Stone for your work, I think it would be very difficult for anyone else to guess what you mean by all your notations.

I see nothing wrong with making your own notes on a passage of text and equations in which you break everything out in a tabular format with displayed equations and no paragraphs of text. Just don't expect anyone else to read it. It is for your own use in organizing your thoughts, and that is all.

If you really want to write things like this in mathematical logic, there are various computer-aided proof systems in which you can write your definitions and theorems in completely symbolic language and feed them into the software, which will check them for you. But I don't know if you would actually find this easier to work with than the text in a book like Apostol's.

Most humans find it much easier to understand proofs written in a natural language (assuming, of course, it is a language that they are fluent in) with logic symbols kept to a minimum. You may find it easy to deal with a proof of four or five lines written in logic symbols, but I suspect it would be quite a different matter with a $100$-page proof. Natural language can be much better at telling you what is going on, while with logic symbols alone you would be lost in the details.

TLDR: Using sentences is necessary. The consensus about this in the mathematics community is overwhelming at least.

The basic idea is that you are not writing your work for yourself. Let me give a simpler example that I find fitting. I am not a native English speaker. It is much easier (for me) to write my thoughts in Greek. But right now I am trying to communicate with you - that means that I must write in a language that is easy for you to understand. That language is English. Most mathematicians will have serious problems understanding what you mean (I tried skimming through your work - it wasn't easy) if you insist on using logic symbols.

Assuming you are not famous and you haven't managed to prove something really "big" that means that your peers will not even bother reading your work.

I really enjoyed reading: https://sites.math.washington.edu/~lee/Writing/writing-proofs.pdf

and laughed a lot with the last line in: https://ocw.mit.edu/courses/mathematics/18-901-introduction-to-topology-fall-2004/assignments/commentsonstyle.pdf